bidden

probably an early Middle English mutual influence or confusion of two older words: The sense in bid farewell is from Old English biddan "to ask, entreat, beg, pray, beseech; order" (class V strong verb, past tense bæd, past participle beden), from Proto-Germanic *bedjanan "to pray, entreat" (source also of German bitten "to ask," attested in Old High German from 8c., also Old Saxon biddian, Old Frisian bidda"ask, request command," Old Norse biðja, Gothic bidjan "request"). This, according to Kluge and Watkins, is from a PIE root *gwhedh- "to ask, pray" (see bead (n.)).

To bid at an auction, meanwhile, is from Old English beodan "offer, proclaim" (class II strong verb; past tense bead, past participle boden), from Proto-Germanic *beudanan "to stretch out, reach out, offer, present," (source also of German bieten "to offer," Old High German biatan, also Old Saxon biodan, Old Frisian biada, Old Norse bjoða, Gothic anabiudan "to command"). This is (with a shift of meaning) from PIE root *bheudh- "be aware, make aware" (source also of bode (v.)).

Middle English biden, from Old English bidan "to stay, continue, live, remain," also "to trust, rely," from Proto-Germanic *bidan "to await" (source also of Old Norse biða, Old Saxon bidan, Old Frisian bidia, Middle Dutch biden, Old High German bitan, Gothic beidan "to wait"), which is of uncertain origin. According to Watkins possibly from PIE root *bheidh- "to trust, confide, persuade" (via notion of "to await trustingly").

Frequent in Middle English (to bide on live was "stay alive;" bide in bay was "stand at bay"). It was preserved in Scotland and northern England, displaced elsewhere by abide in all senses except in the expression bide (one's) time.  "I Bide My Time" is said to be "the motto of the earls of Loudon" in a Scottish context [1806, in a note to "Poetical Words of Sir David Lyndsay"], and it may owe its popularity to Scott's significant use of it in "The Bride of Lammermoor":

Ravenswood, who had assumed the disguise of a sewer upon the occasion, answered, in a stern voice, "I bide my time;" and at the same moment a bull's head, the ancient symbol of death, was placed upon the table. The explosion of the conspiracy took place upon the signal, and the usurper and his followers were put to death.

Related: Bided; biding.

Old English unbedene, "not asked or invited," from un- (1) "not" + bidden. Similar formation in Middle Dutch ongebeden, German ungebeten, Old Norse ubeðinn.