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probably a merger of two old words: The sense in bid farewell is from Old English biddan "to ask, entreat, pray, beseech; order; beg" (class V strong verb, past tense bæd, past participle beden), from Proto-Germanic *bidjan "to pray, entreat" (cf. German bitten "to ask," attested from 8c.), which, 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, p.p. boden), from Proto-Germanic *biudanan "to stretch out, reach out, offer, present," (cf. German bieten "to offer"), from PIE root *bh(e)udh- "to be aware, make aware" (cf. Sanskrit bodhati "is awake, is watchful, observes," buddhah "awakened, enlightened;" Old Church Slavonic bljudo "to observe;" Lithuanian budeti "to be awake;" Old Irish buide "contentment, thanks"). As a noun, 1788, from the verb.