outbid 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]

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.)).