flying mouse-like mammal (order Chiroptera), 1570s, a dialectal alteration of Middle English bakke (early 14c.), which is probably related to Old Swedish natbakka, Old Danish nathbakkæ "night bat," and Old Norse leðrblaka "bat," literally "leather flapper," from Proto-Germanic *blak-, from PIE root *bhlag- "to strike" (see flagellum). If so, the original sense of the animal name likely was "flapper." The shift from -k- to -t- may have come through confusion of bakke with Latin blatta "moth, nocturnal insect."
Old English word for the animal was hreremus, from hreran "to shake" (see rare (adj.2)), and rattle-mouse, an old dialectal word for "bat," is attested from late 16c. Flitter-mouse (1540s) is occasionally used in English (variants flinder-mouse, flicker-mouse) in imitation of German fledermaus "bat," from Old High German fledaron "to flutter."
As a contemptuous term for an old woman, it is perhaps a suggestion of witchcraft (compare fly-by-night), or from bat as "prostitute who plies her trade by night" [Farmer, who calls it "old slang" and finds French equivalent "night swallow" (hirondelle de nuit) "more poetic"].