"fluid drop from the eye," Old English tear "tear, drop, nectar, what is distilled in drops," from earlier teahor, tæhher, from Proto-Germanic *tahr-, *tagr- (source also of Old Norse, Old Frisian tar, Old High German zahar, German Zähre, Gothic tagr "tear"), from PIE *dakru- (source also of Latin lacrima, Old Latin dacrima, Irish der, Welsh deigr, Greek dakryma). To be in tears "weeping" is from 1550s. Tear gas first recorded 1917.
Middle English drope, from Old English dropa "a small, spherical mass of liquid," from Proto-Germanic *drupon (source also of Old Saxon dropo, Old Norse dropi, Dutch drop, Old High German tropfo, German Tropfen (n.)); see drop (v.).
Sense of "minute quantity of anything, least possible amount" is from c. 1200. Meaning "an act of dropping" is from 1630s; of immaterial things (prices, temperatures, etc.) from mid-19c. Meaning "lozenge, hard candy" is 1723, from resemblance in shape. Meaning "secret place where things can be left illicitly and picked up later" is from 1931. Theatrical meaning "painted curtain dropped between scenes to conceal the stage from the audience" is by 1779.
Drop in the bucket (late 14c.) is from Isaiah xl.15 [KJV]. At the drop of a hat "suddenly" is from 1854. To get the drop on "be prepared before one's antagonist" originally was Old West gunslinger slang (1869).