How about Occidentalia or Occident, after the Latin word for "West"? It's not very creative, but there's no guarantee that the new lands would be named after a particular discoverer or get a more "creative" name. My guess is that in English, at least, the continents would be called "North Occident" and "South Occident", since "Occidentalia" just has too many syllables.
If Columbus' career after his initial discoveries had been more successful, one or both continents could easily have ended up being called "Columbia". In OTL, his failure to discover either a passage to the Indies or a major source of gold, plus his horrible record running the earliest Spanish colony on Hispaniola, plus his stubborn adherence to the idea that he was fairly close to Asia, plus his getting shipwrecked for a year on his fourth voyage, plus other factors all contributed to a sharp decline in his influence and popularity. Other explorers (including Vespucci) had already taken over the limelight years before Columbus died.
It's ironic that Amerigo Vespucci was an active explorer only in South America, but ended up having BOTH continents named after him. If there had been another early 16th century explorer who had conducted a whole series of successful and well-publicized explorations of the lands north of the Caribbean, perhaps that explorer would end up with his name attached to the continent. In OTL, some of the early explorers of what would become the eastern coasts of the USA and Canada were John Cabot and his son Sebastian sailing for England, the Corte Real brothers for Portugal, and Verrazano for France. For the Spanish there was Ponce de Leon, Narvaez, then De Soto and Coronado, as well as a couple of other explorers whose names I don't remember right now. If any of them had been either been successful in mapping much of the coastline or starting a successful permanent settlement (or both) we might have South America as "America" and North America could be Cabotia, Realia, Verrazania, Leonia, Coronadia, Sotosia (?) or who knows what else!