Here are the facts. There was an explorer named Alberigo (aka Albericus aka Alberico aka Morinto) Vespucci (aka Vespucius aka Vespuio aka Vespuchy). Vespucci sailed to the Western Hemisphere once under a Spanish flag and the other time for the Portuguese. Later in 1507 a German cartographer named Martin Waldseemuller made a map calling the new world "America."
The main theory is that Walseemuller took the female, Latinized first name of Vespucci and put it on a map.
However, some geographers disagree with this theory. In 1875 Jules Marcou of the Academy of Sciences in Paris proposed the Amerique Mountains of Nicaragua were the origin of the name "America." On Columbus' fourth journey, his fleet was forced to repair along the coast of Nicaragua due to damage by teredo worms. The Indians which were encountered by the crew wore gold jewelry that came from the Amerique Mountains. It is theorized that they told the Spanish they got the gold from "Amerique" and hence that is how "America" first came into being.
Going further Marcou and other likes Jorge Espinosa Estrada, author of Nicaragua, Cuna de America, allege that Walseemuller would only know Vespucci's first name as "Albericus" or "Alberico." Vessucci even published his 1504 pamphlet Mundus Novus under the Latin name Albericus Vesputius. To get America out of that seemed like a stretch to Marcou. Furthermore, it was the custom to name places after last names and not first names. Hence all the places with names like Columbia, Colombia, Colon etc. rather than Cristoforo, Cristobal, etc.
This is the main opposition theory to the commonly accepted origin of America. Catholicgauze is pretty sure the Waldseemuller map's explanation of Amerigo is the correct one, however, having an alternative theory never hurt.
There are other theories however. For fun feel free to read the argument that America should be known as "Cabotia" for John Cabot. This idea is currently being championed by Flat Earth Protestants.