overstay (v.)

mid-15c., steien, 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."

Originally in Middle English also intransitive, "cease going forward, come to a halt," a sense now obsolete. The meaning "remain" (as opposed to "depart, go away") is recorded from 1570s but is implied earlier in late 14c. surname Steyhame. The meaning "reside as a guest for a short period" is from 1550s.

Of things, "remain in place," 1590s. The transitive sense of "render motionless" is by 1620s. The legal sense of "put off, defer, postpone" is from 1520s. Related: Stayed; staying.

As a word of command or injunction, "pause, stop, cease!" by 1580s. Verbal phrase stay put "remain where placed" is recorded by 1834, American English. "To stay put is to keep still, remain in order. A vulgar expression" [Bartlett]. To come to stay "become permanent" is by 1853. A stay-stomach was (1800) "a snack."

word-forming element meaning variously "above; highest; across; higher in power or authority; too much; above normal; outer; beyond in time, too long," from Old English ofer (from PIE root *uper "over"). Over and its Germanic relations were widely used as prefixes, and sometimes could be used with negative force. This is rare in Modern English, but compare Gothic ufarmunnon "to forget," ufar-swaran "to swear falsely;" Old English ofercræft "fraud."

In some of its uses, moreover, over is a movable element, which can be prefixed at will to almost any verb or adjective of suitable sense, as freely as an adjective can be placed before a substantive or an adverb before an adjective. [OED]

Among the old words not now existing are Old English oferlufu (Middle English oferlufe), literally "over-love," hence "excessive or immoderate love." Over- in Middle English also could carry a sense of "too little, below normal," as in over-lyght "of too little weight" (c. 1400), overlitel "too small" (mid-14c.), oversmall (mid-13c.), overshort, etc.