late 14c., "unclaimed property, flotsam, stray animal," from Anglo-French waif (13c., Old French guaif) "ownerless property, something lost;" as an adjective, "not claimed, outcast, abandoned," probably from a Scandinavian source akin to Old Norse veif "waving thing, flag," from Proto-Germanic *waif-, from PIE root *weip- "to turn, vacillate, tremble ecstatically." Compare Medieval Latin waivium "thing thrown away by a thief in flight." A Scottish/northern English parallel form was wavenger (late 15c.).
Meaning "person (especially a child) without home or friends" first attested 1784, from legal phrase waif and stray (1620s), from the adjective in the sense "lost, strayed, homeless." Neglected children being uncommonly thin, the word tended toward this sense. Connotations of "fashionable, small, slender woman" began 1991 with application to childishly slim supermodels such as Kate Moss.
Proto-Indo-European root meaning "to turn, vacillate, tremble ecstatically."
It is the hypothetical source of/evidence for its existence is provided by: Latin vibrare "set in tremulous motion, move quickly to and fro, quiver, tremble, shake," Lithuanian vyburti "to wag" (the tail), Danish vippe, Dutch wippen "to swing," Old English wipan "to wipe."
c. 1300, "deprive of legal protection," from Anglo-French weyver "to abandon, waive" (Old French guever "to abandon, give back"), probably from a Scandinavian source akin to Old Norse veifa "to swing about," from Proto-Germanic *waif-, from PIE root *weip- "to turn, vacillate, tremble ecstatically." In Middle English legal language, used of rights, goods, or women.
If the defendant be a woman, the proceeding is called a waiver; for as women were not sworn to the law by taking the oath of allegiance in the leet (as men anciently were when of the age of twelve years and upwards), they could not properly be outlawed, but were said to be waived, i.e., derelicta, left out, or not regarded. [from section subtitled "Outlawry" in J.J.S. Wharton, "Law-Lexicon, or Dictionary of Jurisprudence," London, 1867]
By 17c. as "to relinquish, forbear to insist on or claim, defer for the present." Related: Waived; waiving.