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"to remain," mid-15c., from Middle French estai-, stem of ester "to stay or stand," from Old French, from Latin stare "to stand" (cf. Italian stare, Spanish estar "to stand, to be"), from PIE root *sta- "to stand" (see stet). Originally "come to a halt;" sense of "remain" is first recorded 1570s.
Noun senses of "appliance for stopping," "period of remaining in a place," and (judicial) "suspension of proceeding" all developed 1525-1550. Stay-at-home (adj.) is from 1806. 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.
"support, prop, brace," 1510s, from Middle French estaie "piece of wood used as a support," perhaps from Frankish *staka "support," from Proto-Germanic *stagaz (cf. Middle Dutch stake "stick," Old English steli "steel" stæg "rope used to support a mast"), from PIE *stak- (see stay (n.2)). If not, then from the root of stay (v.). Stays "laced underbodice" is attested from c.1600.
"strong rope which supports a ship's mast," from Old English stæg, from Proto-Germanic *stagan (cf. Dutch stag, Low German stach, German Stag, Old Norse stag), from PIE *stak-, ultimately an extended form of root *sta- "to stand" (see stet). The verb meaning "secure or steady with stays" is first recorded 1620s.