Since it'll probably come up: America is derived from Amerigo's Latin name, Americus Vespucius. Americus is a masculine name and the continents are all named after women, so Waldseemueller dropped the masculine -us ending and added the feminine -a ending. I suppose we're lucky he didn't prefer Americia, which would (according to my limited understanding of Latin) have worked as well.

Anyway: there's no convention. Baltimore is named after the Lord of Baltimore. Wellington is named after the Duke of Wellington. Stalingrad was named after Ioseb Jughashvili's adopted name. Virginia is named after the the Virgin Queen. Alexander named a city after his horse and a whole bunch after his given name (rather than the name of his dynasty, Argead). The Dominican Republic is named after Dominic de Guzman. Gibraltar was originally named Jabal Tariq after Tariq ibn Zayid. Arlington is named after Henry Bennet's title. There are certainly a lot of places in America named after families, but that's far from a universal standard.

All that being said: I have no earthly idea. The letter that Waldseemueller read was signed Albericus Vespucius, not Americus. The exact rationale is supposed to be contained in Cosmographiae Introductio, but I don't know enough Latin to parse that and I'm not sure I'd trust a translation.

If anyone wants to look at Waldseemueller's 1507 map, Wiki hosts a huge version. He spelled Africa with two f's. He, like Ptolemy, has a rather grand view of what land Ethiopia actually covered.

Fun fact: Waldseemueller tried to change the name later on [there's a lot here, so ctrl+f: withdraw the word "America,"].

There's actually a bit of debate about the origins of the name.

One theory is based around America being named after a region the Native Americans named Amerrique, which the European explorers took with them. This theory was first postulated in the late 1800s and still appears now and again.

Another theory is based around America being named after a wealthy merchant called Richard Amerike.

Both theories are based partly on evidence that show Vescuppi never actually named the land after himself. He may not have had anything to do with the naming at all.

Sources:

http://www.uhmc.sunysb.edu/surgery/america.html

http://www.uhmc.sunysb.edu/surgery/broome.html (old but still interesting)

The reason it was his first name is because America is a continent and the more traditional continental naming scheme was based on first names.

Herodotus gives an account of the names of the three continents (Europe, Libya, and Asia) in his Histories. Each continent's name is the first name of the mythical figure they're named for (at least according to Herodotus). Please excuse the Wikipedia links, but they're the most direct and accessible descriptions of the mythological figures: Europa, Libya, and Asia. While Herodotus' method of naming continents does not strictly speaking represent an actual cartographic rule people followed, see Antarctica and Australia as counter-examples, it seems to have been influential on Waldseemuller when he made his map.

The reason America stuck around as a name is that it was kind of a compromise name. It's worth remembering that America originally meant what we now call South America. The Spanish wanted to call the continent New Spain, while the Portugese obviously had no interest in doing anything of the kind. What to call the new continent became political very quickly and America was the least offensive name that was in fairly widespread circulation so it just sort of stuck.

Edit for more content: I mentioned this in another post here, but the reason Waldseemuller even picked Amerigo Vespucci is that he, mistakenly, believed that Amerigo was the first European to land on the continent of America. While Columbus had indisputably been to Caribbean first what was not as clear was who reached the continental landmass of America first. In hindsight it is pretty clear that it was also Columbus, or at the very least he got there before Amerigo did, but at the time an account of Amerigo's voyages of exploration called Mundus Novus was circulating that argued that he was the first to reach America. Historians now think that this document was not even written by Amerigo when Waldseemuller was making his map it seems that he believed the account of Mundus Novus and so named the continent after Amerigo.

There is also some evidence that Waldseemuller actually recanted his original attribution. There are other maps he made where he abandoned the 'America' label in favor of the slightly more traditional Terra Incognita. However, the exact chronology of Waldseemuller's maps is a tad confused, we don't always have the best records about when he was working, so it is hard to definitively prove that the Terra Incognita map was a later work. There's even some evidence to suggest that the supposed 'First Map with America on it' might actually be the second. That said, there is convincing evidence that in later life he decided to distance himself from the America label, perhaps when it became apparent that Columbus, not Amerigo, was the first to reach South America.

Source: Jerry Brotton A History of the World in 12 Maps While the name suggests this book is a general world history that uses maps as a tool for learning, it's actually a history of cartography from ancient Greece through the present.

Ronald H. Fritze New Worlds: The Great Voyages of Discovery 1400-1600