The origin of the word Jumbo is a bit of a myth mixed in with conjecture and splashed with some showmanship. In essence a good story.
It is thought the original word could have been of African origin. The Kongo words nzamba and nzombo (meaning elephant) are an option as the genesis of the word. Others believe that it could have come from the swahili word “hello” (jambo) or from the Zulu word for a “large parcel”. Basically no one knows where it originally came from but we do know when it became popular and THIS is an interesting story.
In 1860 a young elephant calf was captured somewhere in East Africa after his mother was shot by hunters. He made his way to France and thereafter to the London Zoo. At the London Zoo Jumbo started coming into his own and became the largest elephant the world had ever seen.
The legendary showman P.T. Barnum bought Jumbo from the London Zoo in 1882 and he was shown to the American public.
In London Jumbo had held a similar celebrity to Queen Victoria, so was he revered and known. When in America, thanks to some marketing, his arrival was much anticipated and hence his fame was set.
After his death (he was hit by a train) P.T. Barnum had his remains preserved and these toured with the circus as something of an exhibit.
All of this celebrity, around what was labeled as the largest elephant in the world, meant that thereafter any large thing was referred to as Jumbo.
The reasons the elephant was called Jumbo in the first place are wide. There is a theory that the word originated from mumbo-jumbo and that the guy who was curator of the London Zoo, Abraham Bartlett, liked the word.
Most of this information comes from this page. It has more detail and is worth a read if you have some time.