I will try to supply a few more theories to this discussion if I can: based on the symbolism and hints of the books:
Theories about Arya Stark (or the ones that I could find or develop):
1. Arya is Nissa Nissa and Jon Snow or Gendry Baratheon are Azor Ahai:
We know the story of Azor Ahai and how he created Lightbringer: He made a sword called lightbringer, he worked in it three times, the first time it shattered when it came in contact with the water. On the second time, he used the heart of a LION to cool the iron, but the sword shattered again. On the third time, he called his wife after he had finished the sword, and asked her to uncover her chest: he told her he would always love her and then he drew the sword through her heart. Nissa Nissa's scream shattered one side of the moon, but her courage and her strength passed to the sword and created Lightbringer.
Assuming Ice is the new Lightbringer, it was transformed into two swords, which would represent the second attempt at making Lightbringer, one part of that sword was given to Tommen Baratheon and the other was given to Jaime Lannister, who gave it to Brienne of Tarth. Remember the prophecy: the second time, Azor Ahai tried to cool the iron using a Lion's heart. Does this mean Brienne will kill Jaime using Oathkeeper? Or does it mean Tommen will get killed by Widows wail? In any case, we know that Brienne plans to re-forge Ice, which is why she approaches Gendry. Gendry, just like Azor Ahai, is a blacksmith; and so some hints point at the fact that he may be Azor Ahai: the red commet; the faith of R'hllor; the fire. However there are also theories that point that Jon is actually Azor Ahai revived; Flaming Sword; Melissandre's visions; association with fire. In case that Gendry or Jon were Azor Ahai, the prophecy says that in the times of the Long Night, a warrior shall draw a sword from the fire and this sword will be Lightbringer. Assuming Lightbringer is not the original but a newly forged, better version, and remembering the story of Lightbringer's forge, it is clear this sword will need the heart of a lover to pass through in order to become Lightbringer.
Why can Arya be Nissa Nissa?
Circumstantially, assuming Jon Snow or Gendry Baratheon are Azor Ahai, Arya is the closest to them in a way: We could argue that Jon's Nissa Nissa was Ygritte, yet this would contradict the prophecy: Azor Ahai killed his wife: Jon never killed Ygritte, someone else did.
We know that Arya and Jon love each other more than they love the rest of their siblings: They think about each other constantly, they remember each other constantly. Jon's love for Arya is the only thing apparently capable of making him break his vows and go to rescue her: Not Ygritte, not his brothers Bran and Rickon, not Robb Stark, none of them were strong enough to make him quit. When it comes to Arya, we know the only thing she holds with her is Needle, and that her father's face has almost been erased from her memory, yet she remembers Jon Snow's smile, which is weird. These two love each other. I'd even say their love goes above simple sibling love and enters the world of "platonic lover". This doesn't necessarily mean it is an incestuous relationship, since Jon's parentage is not clear, and doesn't mean this relationship will ever take an erotic turn, but it is clear Arya is Jon's beloved. Even more so than his Wildling wife, Ygritte, kissed by fire.
When it comes to Gendry, we know he's never been really close to anyone, since he's an orphan boy from Flea Bottom and his master apparently "sold him" to the Night's Watch. However, he gets pretty close to Arya. I would say he's platonically in love with her (I have my doubts about the feeling being mutual, however). Just like his father, he seems extremely infatuated with Arya's strength. He bursts into jealous attacks, he's insecure around her (something he isn't in other many situations) and his feelings are clearly represented in the "Maiden of the Tree" song.
There is one problem with this theory, and it is that, well, Arya's heart isn't Nissa Nissa's heart. In fact, we know thanks to the prophecy of the ghost of High Heart that Arya has a dark heart, which would mean that a sword drawn from her heart wouldn't bring light, it would bring darkness. But perhaps this is the spin of the story: Since prophecies can, in Martin's own words, "be misinterpreted easily." Another problem with this theory is that maybe Jon or Gendry aren't Azor Ahai and the real Azor Ahai is Daenerys, in which case the prophecy would not be valid anymore, since Daenerys's Nissa Nissa was Drogo and her "Lightbringer" are the three dragons. Assuming that Daenerys is the right Azor Ahai, here's the next theory:
2. Arya is the Champion of the Great Other and her mission will be to destroy Azor Ahai and its Lightbringer:
This is where the story turns darker: We know too little about Old Valyria, yet in Arya's chapters we find out that the Cult of the Many Faced God was born there amongst slaves and that the first disciple of the Many Faced God started giving the gift to the most desperate slaves and ended up giving the gift to his masters. Maybe it is implied in this simple phrase that it was this Cult the one that caused the Doom of Old Valyria. How does this theory hold up? The famous scene of the Alchemist requesting a book in which is information regarding how to kill dragons. If the Alchemist is, in fact, Jaqen H'ghar, then it means the faceless men are looking for ways to get rid of the Dragons.
Martin has stated that the only way of defeating the White Walkers are dragons. Either the Faceless Men are completely unaware of Winter coming to Westeros, or they don't give a fuck about it, which I don't think is the case. In any case, killing a dragon in this moment means destroying an important weapon meant to protect Westeros against the White Walkers, in other words, serving the White Walkers or the Great Other, which wouldn't be so surprising since the Great Other IS the god of death, that means, another face of the Many Faced God.
Could it be that Arya were, at some point, given the mission of destroying Azor Ahai, in case that Azor Ahai was Daenerys Targaryen, or destroy her "Lightbringer" a.k.a her dragons? In the case she succeeded, Westeros would have no way of defending itself against the attack of the White Walkers, which means Arya would have become the Champion of the Great Other.
Something that seems to point this theory out is the fact that Arya, again, is a dark character. It seems everything she does drives her towards a greater internal darkness. When the Ghost of High Heart sees her she seems to see only tragedy or darkness, or danger in Arya's future. "Go away, dark heart."
The only problem with this theory is that Faceless Men only kill by request. And who would be rich or powerful enough to request the murder of a Dragon? However, many rules of the Faceless are yet to be uncovered, so it could be possible that they have a secret agenda.
3. Arya is the Valonqar, the Younger more Beautiful one, the Avenger of the North, or simply assassinates someone important:
A smaller future for Arya that doesn't necessarily imply a tragedy: we know Arya is a deadly tiny ball of hatred and bloodlust. She wants revenge at any cost. And she is, now more than ever, insanely close to becoming one of the most kick-ass assassins in the world. So this is actually the most logical future for her. We know that at some point in the future she's going to end up killing someone important (by request or without request) and this death will spark a lot of important consequences for the plot.
The three most popular theories are:
3.1. Cersei: The prophecy of Maggie the Frog states that Cersei will be strangled by the Valonqar (Little Brother). This is the reason Cersei is so mental about Tyrion. However, it is probable that Tyrion is not and never was the Valonqar. Valonqar means "little brother", it doesn't mean "your little brother" or "the smallest brother". This opens up a range of new candidates: We know that since Tyrion married Sansa, Starks and Lannisters are actually "brothers in law". Which means that now Cersei has at least four little siblings, all of them very willing to choke the life out of her if given the chance. One of those siblings is Arya. And since Valonqar is a gender-neutral word, it is very possible.
3.2. Tommen: It would seem as if the only thing impeding the great fall from grace of the Lannister House was the fact that Little Tommen is still on the throne and still wielding the power. However we know that it is just a matter of time before he crumbles and then the Lannisters will be smashed either by the Martells or by their newfound enemies the Tyrells. Tommen's fall cannot be simplified more: He needs to die. And there are a few candidates meant for the job, one of them is Arya: A young assassin sent to kill the weak king by coincidence (or maybe not). Why would I refer in this to Arya as the "younger, more beautiful" woman of the prophecy? Because if Cersei holds two things dearly in her life it is her power and her last child. All the rest has been taken away from her: Her daughter, her first son, her beauty (fainting with age), even her brother Jaime. The last thing she has is Tommen: the last representation of her fading power and her last child. Arya's murder of Tommen would indeed take away "everything she holds dear". Arya's coincidental resemblance to Lyanna is a very important part of this: We could argue that Lyanna was Cersei's curse from the moment Cersei became queen. A constant phantom in Robert Baratheon's head, the memory that Lyanna got Robert's AND Rhaegar's love, two things Cersei wanted and could never get. Lyanna has certainly haunted Cersei. A second Lyanna killing her son would drive her mad.
3.3. Ramsay Bolton: At some moment in time someone will have to kill this motherfucker. And although we would love Reek to do it and take revenge for his lost little friend (and by this I mean his penis), he's very unlikely to be able to do so thanks to the traumas of the psychological and physical torture Ramsay put him through. It gets also very clear when Reek abandons Winterfell, that the killer must be a northerner or a northern creature, since it is "the North" that "remembers", not the Iron Islands. In case it is not Jon, nor Rickon, nor Sansa (Bran wouldn't kill him, he's made his hippie vows to a talking tree), the candidate is Arya. This gets enhanced by the fact that Ramsay is actually married to her. Even though it is not the real Arya Stark, the marriage is apparently valid. Since Arya is the Bride of the Stranger, however, this indirectly means Ramsay is going to die one way or another, since nobody fucks with Death's girl. Now you could argue that it is not real Arya the one he's married to, but just as Jaqen would say "One name is enough". In case Arya came to Winterfell and killed her sweet loving husband, the Capitol of the North would be in the hands of a Stark again, making Arya a Queen in the North (at least until her more educated in politics sister came, and then they could run it together like the badasses they are and throw food at each other like old times (although they probably shouldn't waste food since Winter is Coming)).
3.4. Gregor Clegane: Gregor Clegane died due to the poison of Oberyn's spear in Feast for Crows. However, the evidence points to the fact that Ser Robert Strong may be Gregor Clegane, revived by Qyburn and put inside the Kingsguard as Cersei's last plan of protection. Because as long as Ser Robert Strong lives, Cersei and her family are technically safe. This is confirmed by Kevan Lannister. Qyburn mentioned he was working on "creating" an unbeatable champion for Cersei. In fact, the armor he ordered for that champion was so thick and heavy that the blacksmith stated no man was strong enough to wear it. It is required to have an incredible soldier or warrior in order to destroy Robert Strong. Even when Gregor was alive, no one could defeat him. The theories about the warrior that would be able to defeat this monster mostly point towards the Hound. But why could it be speculated that Arya is the Champion? During Bran's first visions he sees a smash of prophecies about his brothers, sisters and parents. One of these prophecies involves his two sisters: Sansa crying until she fell asleep, Arya watching and holding the hatred inside her heart. Behind them Bran could see a blond knight in shining armor (Jaime) and a dog (Sandor) both of them too far away from both girls to reach them. Over the girls, however, there was a shadow, the shadow of a soldier. When he opened the visor of his helmet there was no head, only a black void, from which blood streamed down. Although we could argue that this soldier is not necessarily Robert Strong but an allegory to the Stranger or maybe to the Titan of Braavos, it seems kind of fishy that this soldier, without a head, has so much in common with Robert Strong: First, if Robert Strong is in reality Gregor Clegane, Gregor's head has been sent to Dorne, and the lack of Robert's head could explain why he doesn't eat, speak or sleep. He's a headless, animated body. We know that the Hound and Jaime were the greatest warriors once and that they could have been able to defeat the Mountain, yet they never did it. Jaime was never interested, and later he lost his hand. Sandor saw no point in killing his brother. But what is the connection between Robert Strong, Arya and Sansa? Why are they together in Bran's vision? Maybe because they ARE the ones who will destroy the warrior that could never have been destroyed. Since Sansa cannot wield a sword, it would make sense that if Robert is to be defeated, it would be by Arya's hand. And yet Sansa will have a very important part to play in the whole scheme as well. Prophecy also states that she will one day defeat a savage giant in a castle of ice. This savage giant could be Littlefinger, (although it is unlikely, since Littlefinger is no savage), or Robert Strong. There's another side of this theory and a problem with it, which I'll explain in point 5:
5. Robert Strong's theory's dark side:
Bran's vision could also point out at the fact that maybe Sansa and Arya will die by Robert Strong's hand. We know that Sandor and Jaime have tried to save both girls: The Hound tried to persuade Sansa to run with him. He wanted to take Arya to safety by her aunt's side (for the money, but that doesn't matter). Jaime swore to Lady Catelyn that he would protect her daughters. He never found them, but he commended the task to Brienne, unable to let go of his vow. Both the Hound and Jaime failed. Jaime arrived too late in King's Landing, when both girls had fled the city, Sandor had to leave without Sansa and was left to die by Arya. This is the symbolism of both Knights in Bran's vision, both hovering over the girls, but too far away from them to even reach them. The only one who seems to hover over them close enough is this shadow of the headless soldier. Could this shadow be Robert Strong? In which case we know that, since Robert Strong is a protector of the Queen, not of the Starks, it is actually a premise of death for both girls.
What's the problem with this theory? That the giant does not necessarily have to be Robert Strong. There is a possibility that this Giant is actually the Titan of Braavos, which would make sense given Arya's and Sansa's story arc. Arya has been followed around and protected by braavosi people since she came to King's Landing: Syrio was braavosi, Jaqen claimed to be from Lorath, but if he belonged to the Faceless Men he had definitely lived in Braavos. Arya was shipped to Braavos thanks to him, she had a guardian angel looking over her in Braavos, which would explain the headless knight hovering over her: The Titan of Braavos. What does Sansa have to do with Braavos? Petyr Baelish's grandfather was a braavosi sell-sword. His son took the STONE HEAD of the titan of Braavos as his sigil. Sansa has also been taken care, followed and protected by Petyr for a long time. And according to the theory of Petyr's master plan, he has some serious business in Braavos. So, the soldier in Bran's vision does not necessarily mean Robert Strong, it could mean the Titan of Braavos doing what Jaime and Sandor could not: keeping the girls safe.