I was taught that America was named after Amerigo Vespucci, but here are some variant theories, copied from wikipedia.org. Go to that page for live links, or to read the full post.:

The earliest known use of the name America for this particular landmass dates from April 25, 1507. It appears first on a small globe map with twelve time zones, and then a large wall map created by the German cartographer Martin Waldseemüller in Saint-Dié-des-Vosges in France. An accompanying book, Cosmographiae Introductio, explains that the name was derived from the Latinized version of the Florentine explorer Amerigo Vespucci's name, Americus Vespucius, in its feminine form, America, as the other continents all have feminine names.

Vespucci's role in the naming issue, like his exploratory activity, is unclear. Some sources say that he was unaware of the widespread use of his name to refer to the new landmass. Waldseemüller may have been misled by the Soderini Letter, claimed by some to be a forgery, which implies that it was discovered first by Amerigo Vespucci. Christopher Columbus, who had first brought the region's existence to the attention of Renaissance era voyagers, had died in 1506 (believing, to the end, that he had discovered and colonized the Indies he had set out looking for and could not protest Waldseemüller's decision.

An alternate proposal, first advanced by Jules Marcou in 1875 and later recounted by novelist Jan Carew, is that the name America derives from the district of Amerrique in Nicaragua. The gold-rich district of Amerrique was purportedly visited by both Vespucci and Columbus, for whom the name became synonymous with gold. According to Marcou, Vespucci later applied the name to the New World, and even changed the spelling of his own name from Alberigo to Amerigo to reflect the importance of the discovery.

Another theory, first proposed by a Bristol antiquary and naturalist, Alfred Hudd, in 1908 was that America is derived from Richard Amerike (Richard ap Meurig), a Welsh merchant from Bristol, who is believed to have financed John Cabot's voyage of discovery from England to Newfoundland in 1497 as found in some documents from Westminster Abbey a few decades ago. Supposedly, Bristol fishermen had been visiting the coast of North America for at least a century before Columbus' voyage and Waldseemüller's maps are alleged to incorporate information from the early English journeys to North America. The theory holds that a variant of Amerike's name appeared on an early English map (of which no copies survive) and that this was the true inspiration for Waldseemüller.