1580s, in logic and grammar, in the latter "division of one syllable into two," from Latin, from Greek dialysis "dissolution, separation" (used of the disbanding of troops, a divorce, etc.), from dialyein "dissolve, separate," from dia "apart" + lyein "loosen" (from PIE root *leu- "to loosen, divide, cut apart").
Chemistry sense of "process by which particles are selectively removed from a liquid by consequence of their differing capacity to pass through a membrane into another liquid" is from 1861; the specific sense in medicine, "process of blood purification by allowing it to pass through a membrane" is by 1914. Related: Dialytic. The verb dialyze was formed from the noun on the model of analyze, etc.