c.1300, "performance of an act set forth by a priest or other Church authority to atone for sin," from L. satisfactionem (nom. satisfactio) "a satisfying of a creditor," from satisfacere (see satisfy). Sense of "contentment" first recorded late 14c.; not common before 16c.
The modern verb is a merger of two old words. The sense in bid farewell is from O.E. biddan "to ask, entreat, pray, beseech; order; beg" (class V strong verb, past tense będ, past participle beden), from P.Gmc. *bidjan "to pray, entreat" (cf. Ger. bitten "to ask," O.H.G., 8c.), which, according to Kluge and Watkins is from a PIE base *gwhedh- "to ask, pray" (see bead). A bid at an auction, meanwhile, is from O.E. beodan "offer, proclaim" (class II strong verb; p.t. bead, p.p. boden), from P.Gmc. *biudanan "to stretch out, reach out, offer, present," (cf. Ger. bieten "to offer"), from PIE base *bh(e)udh- "to be aware, make aware" (cf. Skt. bodhati "is awake, is watchful, observes," buddhah "awakened, enlightened;" O.C.S. bljudo "to observe;" Lith. budeti "to be awake;" O.Ir. buide "contentment, thanks").