I'm looking at the paper "On the theory of local rings" by Chevalley (Annals of Math. 44 (1943)). In this paper he explains how to localize at a multiplicative set $S$ of non-zero divisors, and calls this the ring of quotients of the set $S$.
Of course none of these references quite address the origin of the term localization at $S$, but (based on my prior preconceptions, and bolstered by having looked at these two papers) I am fairly confident that it was indeed motivated by algebraic geometry.