To waive something is to voluntarily give up the right to that thing or to refrain from enforcing a rule or regulation, and a waif is an orphaned or abandoned child. But the two words are very much related etymologically. Both date to the thirteenth century and come into English from Norman French. The ultimate root is probably Scandinavian in origin, as there are Old Norse cognates, but the history of the word before the Norman Conquest is quite fuzzy.
The Oxford English Dictionary only has two citations of the verb to waive from the thirteenth century, both referring to the exile or outlawing of a person. One is in English, found in Robert of Gloucester’s Chronicle for the year 1297; the other is in Anglo-Latin. The verb became more common and broadened in meaning beginning in the late fourteenth century. In addition to the sense of outlawing a person, it could mean to refuse or decline something, to disobey a command, to resist temptation, to give up or yield something, or to grant or convey something.
The noun waif appears before the verb, or at least it is far more common in thirteenth-century texts. It was originally a legal term referring to ownerless or abandoned property. This remained the only sense of the noun until the seventeenth century, when it began to be applied to people who existed on the outskirts of society. John Donne, for example, writes in his Devotions 13 in 1624, referring to person who does not know Christ:
What a Wayue, and Stray is that Man, that hath not thy Markes vpon him?
And it isn’t until the eighteenth century that the modern sense of an uncared for or abandoned person, especially a child, appears.
Waive and waif have a history distinct from the verbs to wave and to waver, at least over the course of the last thousand years. They do, however, probably share a common Germanic root.
Middle English Dictionary, 2001, s. v. weiven, v(1); weif, (n.)
Oxford English Dictionary, second edition, 1989, s. v. waive, v.1; waif, n.1 and adj.
Copyright 1997-2017, by David Wilton