I think people tend to forget just how much land that is - even going to Alta California would be more than half of Mexico, and more land than the US had already gained with Louisiana. Yeah, it may have been underpopulated - but so was the rest of the western US at the time. What would prevent a post-Recovery Mexico from pulling the same stunt as the US with Texas, and sending settlers up into lands the US only nominally ruled? Not to mention that the US couldn't even agree on how much land it wanted to take, with the whole slavery debate going on and all.
That being said, if it did happen, names would probably be rather boring. Short stuff that comes out well in English would probably be preserved - see how they kept the French name for Louisiana and the Spanish one for Florida - but otherwise, stuff will probably get renamed for early American settlers or cities back east, kind of how cities in the Louisiana territories got named IOTL. Or native-sounding stuff. The US tried to use Idaho, god forbid, several times for a state's name. Do you know what Idaho means? It doesn't mean anything, it just "sounded native"
OOC: The Idaho thing is contested origin, but I thought it was still a fair point.