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."