outstay v.

in Old English a common prefix with nouns, adjectives, adverbs, and verbs, "out, outward, outer; forth, away," from out (adv.). The use was even more common in Middle English, and also with the senses "outer, outside, on the outside, from without, external, externally; apart; greatly, extremely; completely, thoroughly, to completion." Other senses of out that extended into the use as a prefix include "beyond the surface or limits; to the utmost degree; to an explicit resolution."

In composition out has either its ordinary adverbial sense, as in outcast, outcome, outlook, etc., or a prepositional force, as in outdoors, or forms transitive verbs denoting a going beyond or surpassing of the object of the verb, in doing the act expressed by the word to which it is prefixed, as in outrun, outshine, outvenom, etc. In the last use especially out may be used with almost any noun or verb. [Century Dictionary]

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."