I don't know much about Vespucci, only that he was Italian and the Americas were named after him. Is there a specific thing he did to have the new world named after him?
He was the first person to declare that the American continent was a "new world" and not part of Asia.
He was also (probably erroneously) sometimes believed to have landed on the mainland of the Continent before Columbus did.
His account of America as a 'new world' was known to the map maker Waldseemueller, who labeled the continent "America" on a map he made in 1507.
This map became very popular and was responsible for spreading the name.
You can get a lot more detail from early posts on this sub-reddit. Here is one post with much more:
https://www.reddit.com/r/AskHistorians/comments/3c8wa6/when_naming_america_why_do_they_use_amerigo/