Most people know that America is named after Amerigo Vespucci, but few know why. Two myths about Vespucci are common. The first is that Vespucci was a fraud who never traveled to the New World. The second is that he was the first European to set foot on mainland America. Both are untrue.
Vespucci made two trips to the New World as a ship's navigator, the first in 1499. Then in 1503 and 1504 he published two letters he had written to Lorenzo de Medici about his voyages. In the letters he put forward the idea that what Columbus had discovered was not in fact a new route to Asia, but rather a new continent. Vespucci also published the first letter under the title Novus Mundus, or New World, thereby coining that phrase. The letters were a media hit (but whether their popularity was because of his innovative navigational theories or his description of the --- lives of American Indians is a question), and Vespucci became a celebrity.
In 1507, the cartographer Martin Waldseemueller published a map in his book Cosmographiae Introductio that designated the new world as America and the name was coined. Like many Italians of his era, Vespucci used the Latinized version of his first name, Americus, in formal writing. Waldseemueller used the feminine form of the name, America, because the names of Europe and Asia were also in the feminine.
So, America is named after the man who first recognized that it was a new continent and not just a part of Asia. Rather fitting actually, at least from a European perspective.
There is a legend, common in Britain, that America is actually named after a 15th century Bristol merchant named Richard Ameryk, who had some tenuous and vague connection with Cabot's voyages of exploration (exactly what role he played is not known, but it was probably minor and connected with financing the voyages). While Ameryk did exist, there is no evidence to indicate the New World continents are named after him. Waldseemueller's notes clearly indicate that he named the continents after Vespucci and most of the claims by supporters of the Ameryk hypothesis are not supported by anything other than a coincidence in spelling and a fierce, English patriotism that wishes the tale were true.