"muddy place," Old English sloh "soft, muddy ground," of uncertain origin, perhaps from Proto-Germanic *slokhaz. Figurative use, e.g. of moral sunkenness or Bunyan's "Slough of Despond," attested from mid-13c.
"to cast off" (as the skin of a snake or other animal), 1720, originally of diseased tissue, from Middle English noun slughe, slouh "shed skin of a snake" (c.1300), probably related to Old Saxon sluk "skin of a snake," Middle High German sluch "snakeskin, wineskin," Middle Low German slu "husk, peel, skin," German Schlauch "wineskin;" from Proto-Germanic *sluk-, of uncertain origin, perhaps from PIE root *sleug- "to glide." Related: Sloughed; sloughing.