From Encarta Encyclopaedia:
The word first appeared in Cosmographiae Introductio (Introduction to Cosmography), edited and published in 1507 by the German cartographer Martin Waldseemüller. The name was derived from Americus, the Latinized given name of the Italian navigator Amerigo Vespucci, whose expeditions to the New World are described in the work. As used by Waldseemüller, America specifically referred to the lands discovered by Christopher Columbus, Vespucci, and other early explorers of the Caribbean and the north-eastern coast of South America. The Flemish geographer Gerardus Mercator first used the word to indicate all the western hemisphere on a map of the world published in 1538.
It doesn’t mention why Americus was used instead of Vespucci, however. This link does:
A more interesting question is why the cartographer Martin Waldseemueller in 1507 named the New World (actually, just South America) America rather than Vespucciland—although I guess to ask the question is to answer it. Amerigo’s first name was a lot more euphonious than his last name, and (no small matter) could be latinized into a word that started and ended with the letter A, just like Asia and Africa before it. Also, unlike Christopher Columbus, invariably referred to by his last name, Vespucci was one of those people known in his own lifetime mostly by his first.
This link explains why it’s not Columbia/Colombia.