Other possibilities abound, such as Columbus referring to a mountain range in Nicaragua called Amerique. However, I think this would be unlikely.
One of the more likely ideas these days comes from an Englishman named Rodney Broome. In a book called "Amerike The Briton Who Gave America Its Name", Broome tells the history of a man named Richard Amerike, a wealthy aristocrat and merchant who funded many expeditions for the purpose of finding new fishing grounds. It's thought that many of these new fishing grounds were in the area of Newfoundland.
There had long been a suspicion that fishing ships in search of cod were regularly crossing the Atlantic from Bristol to Newfoundland before Columbus' first voyage. Bristol merchants bought salt cod from Iceland until 1475, when the King of Denmark stopped the trade. In 1479 four Bristol merchants received a royal charter to find another source of fish. Records discovered in 1955 suggest that from 1480, twelve years before Columbus, English fishermen may have established a facility for processing fish on the Newfoundland coast. In 1960 trading records were discovered that indicated that Richard Amerike was involved in this business. A letter from around 1481 suggests that Amerike shipped salt (for salting fish) to these men at a place they had named Brassyle. The letter also states that they had many names for headlands and harbours. Rodney Broome and others suggest that one of these names may have been "America".It's also known that Amerike funded many of John Cabot's voyages and the thinking here is that John Cabot actually discovered America and not Columbus. It's theorized that Columbus was actually using charts supplied by the English merchants.
Cabot is known to have produced maps of the coast from Maine to Newfoundland, though none have survived. He named an island off Newfoundland St. John's. Copies of these maps were sent to Spain by John Day, where Christopher Columbus and Amerigo Vespucci would have seen them. The theory suggests that Cabot may have written the name America (or similar) on his maps, but no extant maps are available to prove this assertion.Kinda makes Columbus day a little more lackluster, no? It's interesting to see the politics of discovery at work. Columbus is the more celebrated, but because he didn't figure out that this was a new continent he was exploring, he never made an effort to rename what he had found until it was too late. Finders keepers, I guess.
Vespucci sailed to South America and the Caribbean with Alonso de Ojeda (Hojeda) in 1499 and Gonçalo Coelho in 1501 and became convinced that these were new lands, not Asia as Columbus believed. Martin Waldseemüller, a German map-maker, published a world map in 1507 using Vespucci's previously published letters. The theory suggests that Waldseemüller assumed that the "America" that Vespucci used was derived from his first name. Waldseemüller provided an explanation of this assumption as an attachment to the map. Vespucci himself never stated that this was the case. There were immediate protests from Columbus' supporters to get the continent renamed for Columbus, but attempts were unsuccessful, since 1,000 copies of the map were already in circulation. On later maps Waldseemüller substituted the words "Terra Incognita," but it was too late; the name America was now firmly associated with the entire northern and southern continent across the Atlantic from Europe.