Etymology
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lung (n.)

human or animal respiratory organ, c. 1300, from Old English lungen (plural), from Proto-Germanic *lunganjo- (source also of Old Norse lunge, Old Frisian lungen, Middle Dutch longhe, Dutch long, Old High German lungun, German lunge "lung"), literally "the light organ," from PIE root *legwh- "not heavy, having little weight" (source also of Russian lëgkij, Polish lekki "light;" Russian lëgkoje "lung").

So called perhaps because in a cook pot lungs of a slaughtered animal float, while the heart, liver, etc., do not. Compare Portuguese leve "lung," from Latin levis "light;" Irish scaman "lungs," from scaman "light;" Welsh ysgyfaint "lungs," from ysgafn "light." See also lights, pulmonary. Lung cancer is attested from 1882. Lung-power "strength of voice" is from 1900.

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*legwh- 
Proto-Indo-European root meaning "not heavy, having little weight."

It forms all or part of: alleviate; alleviation; alto-rilievo; carnival; elevate; elevation; elevator; leaven; legerdemain; leprechaun; Levant; levator; levee; lever; levity; levy (v.) "to raise or collect;" light (adj.1) "not heavy, having little weight;" lighter (n.1) "type of barge used in unloading;" lung; relevance; relevant; releve; relief; relieve.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit laghuh "quick, small;" Greek elakhys "small," elaphros "light;" Latin levare "to raise," levis "light in weight, not heavy;" Old Church Slavonic liguku, Russian lëgkij, Polish lekki, Lithuanian lengvas "light in weight;" Old Irish lu "small," laigiu "smaller, worse;" Gothic leihts, Old English leoht "not heavy, light in weight."
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allay (v.)

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.

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