at drop of hat

Old English dropa "a drop of liquid," from Proto-Germanic *drupon (cf. Old Saxon dropo, Old Norse dropi, Dutch drop, Old High German tropfo, German Tropfen (n.)), from PIE *dhreu-.

Meaning "an act of dropping" is from 1630s; of immaterial things (prices, temperatures, etc.) from mid-19c. Meaning "lozenge, hard candy" is 1723. Meaning "secret place where things can be left illicitly and picked up later" is from 1931.

Drop in the bucket (late 14c.) is from Isa. ix:15 [KJV]. At the drop of a hat "suddenly" is from 1854; drop-in "casual visit" is 1819; drop-kick is 1857. To get the drop on someone originally was Old West gunslinger slang (1869).