mid-15c., "cease going forward, come to a halt," also (transitive) "detain, hold back," from Old French estai-, stem of estare "to stay or stand," from Latin stare "to stand, stand still, remain standing; be upright, be erect; stand firm, stand in battle; abide; be unmovable; be motionless; remain, tarry, linger; take a side," (source also of Italian stare, Spanish estar "to stand, to be"), from PIE root *sta- "to stand, make or be firm." Sense of "remain" is first recorded 1570s; that of "reside as a guest for a short period" is from 1550s. Related: Stayed; staying.
Of things, "remain in place," 1590s. Stay put is first recorded 1843, American English. "To stay put is to keep still, remain in order. A vulgar expression" [Bartlett]. Phrase stay the course is originally (1885) in reference to horses holding out till the end of a race. Stay-stomach was (1800) "a snack."
"support, prop, brace," 1510s, from French estaie "piece of wood used as a support," Old French estaie "prop, support," perhaps from Frankish *staka "support" or some other Germanic word, from Proto-Germanic *stagaz (source also of Middle Dutch stake "stick," Old English steli "steel," stæg "rope used to support a mast"), from PIE *stak- "to stand, place" (see stay (n.2)). In some uses from stay (v.2).
"strong rope which supports a ship's mast," from Old English stæg "rope used to support a mast," from Proto-Germanic *stagaz (source also of Dutch stag, Low German stach, German Stag, Old Norse stag "stay of a ship"), from PIE *stak- "to stand, place," perhaps ultimately an extended form of root *sta- "to stand, make or be firm."
"support, sustain," early 15c., from French estayer (Modern French étayer), originally in nautical use, "secure by stays," from estaie (see stay (n.1)). The nautical sense in English is from 1620s. Related: Stayed; staying.
1520s, "delay, postponement, period of remaining in a place," from stay (v.1). Meaning "action of stoppage, appliance for stopping" is 1530s; that of "suspension of judicial proceedings" is from 1540s.