Middle English alegen, from Old English alecgan "to put, place, put down; remit, give up, suppress, abolish; diminish, lessen," from a- "down, aside" (see a- (1)) + lecgan "to lay" (see lay (v.)). A common Germanic compound (cognates: Gothic uslagjan "lay down," Old High German irleccan, German erlegen "to bring down").
Early Middle English pronunciations of -y- and -g- were not always distinct, and the word was confused in Middle English with various senses of Romanic-derived alloy (v.) and especially a now-obsolete verb allege "to alleviate, lighten" (from Latin alleviare, from ad "to" + levis "light" in weight; from PIE root *legwh- "not heavy, having little weight").
Amid the overlapping of meanings that thus arose, there was developed a perplexing network of uses of allay and allege, that belong entirely to no one of the original vbs., but combine the senses of two or more of them. [OED]
Hence senses "lighten, alleviate; mix, temper, weaken." The confusion with the Latin words probably also accounts for the unetymological double -l-, attested from 17c. Related: Allayed; allaying.