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).
"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.