Goes all the way back to at least Proto-Germanic *finnaz for Sami, possibly from "to travel by foot, walk". Not terribly unrealistic given the semi-nomadic lifestyle of Sami, Sami and Suomi might just be the same word via general Uralic relatedness (and sound the same to Germanic ears anyway) also what's now considered Finns was back then a rough assemblage of tribes consisting of multiple Uralic immigration waves known to Swedes as "Forest Finns", as supposed to the herding Finns (ie. nowadays Sami) further north.

Which I guess makes Estonians Swamp Finns (in so far as Finland-Finns aren't claiming that one for themselves with their swamp forests) and Hungarians... Paprika Finns.