The story
From Ancient Greek αἱμο- (haimo-, “blood”) + διάλυσις (diálusis, “separation, dissolution”), from διαλύω (dialúō, “I separate”), from διά (diá, “through”) + λύω (lúō, “I loose”). By surface analysis, hemo- + dialysis.
Etymology from Wiktionary, CC BY-SA 4.0.