Originally, the term “Stalemate” was just stale, the mate part was added, possibly originally in jest, because a stalemate, like a checkmate, serves to end the game.
Unlike checkmate, and chess itself, both of which come from Arabic, stale seems to have originated in the French estaler (haler), one of the meanings of which is “to fix,” in the sense of “to set in place.”
The latter is related to the German stellen, “to place,” which, in turn, has been traced to the Teutonic stal, “a fixed place.”