"a vigil beside a dying person," 1865, from death + watch (n.) "a watching." The death-watch beetle (1660s) inhabits houses, makes a ticking noise like a pocket-watch, and was superstitiously supposed to portend death.
FEW ears have escaped the noise of the death-watch, that is, the little clicking sound heard often in many rooms, somewhat resembling that of a watch; and this is conceived to be of an evil omen or prediction of some person's death: wherein notwithstanding there is nothing of rational presage or just cause of terror unto melancholy and meticulous heads. For this noise is made by a little sheathwinged grey insect, found often in wainscot benches and wood-work in the summer. [Browne, "Vulgar Errors"]