Entries linking to bade
probably an early Middle English mutual influence or confusion of two old 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.)).