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 $S$.

There is no question that Chevalley was motivated by algebraic geometry.

The paper "Generalized semi-local rings", by Zariski (Summa Brasiliensis Math. 1 (1946)) attributes the theory of local rings to Krull (in a paper called "Dimensionstheorie in Stellenringen", Crelle 179 (1938), which I don't have a copy of at hand) and Chevalley (in the above mentioned paper), so it seems that the Chevalley reference above is a reasonable guide to the situation.

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.