If a potato has more personality than the protagonist of a story, the writer is doing it wrong.
The best stories are driven by their characters. The best characters aren't two-dimensional constructs of excessive verbiage, but a reflection and amplification of the myriad facets of humanity. Characters must resonate with readers, acting, talking and thinking the way people in their situation would do. Shaped by their background, genes, personality and networks, these characters take on a life of their own, and in doing so become distinct people in their own.
Character creation is complex. The more complicated and technical a person is, the more likely a creator will make a mistake somewhere, creating a false note that jars a reader's sensibilities. If there are too many mistakes, or if the mistake were too serious, the reader would drop the story there and then. Creators must give their all when building characters; at the very least they must try their best. But the ones who reach for the potato protagonist don't even try.
The potato protagonist is as blank as a potato and has the personality of one. Everything about him is dull, flat and humdrum. His skills, backgrounds and talents don't matter; his core is empty, his thoughts and behaviours utterly predictable by anyone familiar with standard storytelling tropes. There is nothing about him that makes him stand out from other protagonists, nothing that draws and retain the reader's attention.
A classic example of the trop is Ichijo Raku of Nisekoi. Ichijo is allegedly the son of a yakuza family at odds with a rival gang. To prevent a gang war, Ichijo must pretend to date the daughter of the rival gang boss, Chitoge Kirisaki, during his high school years. This could have been a fascinating setup, but the creator wasted the potential of the main character.
(Unmarked spoilers ahead!)
Ichijo's background has minimal influence on him. He is supposed to be the heir of a nation-spanning yakuza group, with an army of servants at his command and a fortune to his name. But from the get-go he spurns the notion of inheriting the group, and insists on getting an ordinary job in the real world. This is a thin excuse to explain how and why he goes to a regular school, but this falls flat.
Someone who grew up surrounded by wealth, luxury and (allegedly) murderous yakuza acts, talks and thinks differently from a regular person. He would have a cavalier attitude towards money and possessions, yet he would act and talk with grace and refinement. He would be mindful to act in a way that would not bring dishonour to his family, because the yakuza are allowed to exist only because of the goodwill they have built in their community, and because in the underworld, careless words leads to deadly violence. He would have been groomed to study people, keep track of favours and relationships, network with the children of the rich and powerful, and influence people. Depending on how violent the underworld is at that time, he would also have trained in martial arts and learned how to use illegal weapons.
Instead, Ichijou comes off as an ordinary high school boy. Indeed, his background is almost never referenced until a story arc demands it. For much of the story, you can replace 'scion of a powerful yakuza family' with 'ordinary high school student' and it would not affect him one bit. Ichijou fits the mold of Bland Shounen Harem Protagonist to a T. Nothing about Ichijou makes him stand out from any of the thousands of high school student protagonists out there...except for his utter inability to notice how the girls around him feel about him until the manga draws to a close.
Fundamentally, the potato protagonist is not meant to uphold a story. He exists to solve a marketing problem.
The primary target audience of shounen anime and manga are Japanese high school boys. The easiest way to reach out to them is to have a protagonist that vaguely reflects them and allows them to project themselves into the character. By granting the main character the personality of a potato, the audience has an empty vessel to pour their own unique selves into.
The same applies to other audiences of other categories. Want to write a trashy romance story for women? Create a blank ordinary everywoman. Drawing a shoujo manga? Have a fluffy emotional girl as the protagonist and a cool, handsome boy as the love interest. Writing a men's action adventure novel? Make the protagonist a cold killer and play up the guts and gore.
The Potato Protagonist is easy, but writing is about truth, and most of the time, when employed this trope does not reflect the truth of the world. High school students do not embark on grand adventures; at least not without coming through unchanged. People do not exist to reflect the quirks and desires of other people.
And for characters to be realistic, they must pass as people.
The Potato Protagonist Done Right
(Mass Effect 3 wallpaper, http://www.hdwallpapers.in/female_shepard_in_mass_effect_3-wallpapers.html)
The point of Tired Tropes is not to deride a targeted trope, but to see how it can be employed effectively. And even potato protagonists can be redeemed.
Potatoes are bland lumps. They absorb the flavor of the foods, spices and oils they are cooked with. They can be steamed, fried, boiled, stewed, roasted, grilled or microwaved. They can be cooked as is or cut up into different shapes. This essential malleability is key to properly understanding this trope.
Potato Protagonists lend themselves well to choice-driven games, especially role playing games. The point of such games is to allow the player to shape his experience in the game world with the protagonist as his vehicle. As such, a protagonist without any unwanted baggage is excellent -- the player is free to act however he likes within the confines of the game, without having to experience dissonance between a protagonist's actions and his supposed background.
Where the protagonist does have a backstory, the intelligent developer would find ways to integrate that backstory into the overall choice mechanic to create a deeper gameplay experience. In Mass Effect, the player is free to customise his own Commander Shepherd , and can choose between three separate backgrounds. But these backgrounds exist independently of the player's choice. If the player wants to play a Shepherd who ordered a massacre but later regretted his actions and is trying to be a better person, he can. if the player wants to play a Shepherd who survived a slave raid by hostile aliens, propelling him to become a ferocious war hero and twisting him into a ruthless xenophobe, he can. In games that allow players to deeply customise their experiences and see themselves as active participants in the story events, the potato protagonist is unmatched.
In print media, a potato protagonist is also acceptable...if he does not remain one. Events change people. Stories change characters. The reader must be able to compare a character at the beginning of the story with his future self and see how much he has changed. A potato protagonist facilitates character development, since there is no fear of violating established background or character regression. This also has the effect of making character development appear more obvious to the reader.
An example of this is Rosario + Vampire. Aono Tsukune is an ordinary high school boy who accidentally gets enrolled in a high school for monsters. Predatory monsters who feed on humans and who are learning how to blend into human society, starting with magic that makes them appear human. His innate humanity attracts the attention of a group of monster girls, leading to harem hijinks.
Not.
The story begins as a generic Monster of the Week manga. Then the creator delves into each character's personal life, creating opportunities for drama and character bonding and deconstructing the Unwanted Harem trope. Soon, villains appear, threatening the fragile peace between humans and monsters, and targeting Tsukune and his friends. Tsukune, in turn, resolves to help his newfound friends and love interest, and embarks special training to grow stronger. At the start of the series, Tsukune is a high school boy well over his head, desperately trying not to be unmasked as an actual human; by the final arc of the second season, Tsukune stands alongside his friends to save humanity from a monster terrorist organisation.
Personally, I dislike potato protagonists. Such characters hold little appeal to me, and it takes a great deal of work for me to continue putting up with them longer than an hour. That is usually because they aren't understood and employed properly. But done right, they can become icons in their own right. Case in point: Commander Shepherd.
The potato protagonist is the quintessential blank slate. In choice-driven games, he is a vessel for the player to shape his experiences. In fixed stories, he has maximum potential for evolution and development, pushing the story to greater heights. In both cases, the protagonist changes into someone better.
A potato protagonist is not enjoyed raw; he must be prepared and cooked through the events of the story. Or, like a raw potato, he could poison the reader and turn off the reader permanently.