Long chain of origins: Old English lungen, from Proto-Germanic lunganjo (enlargement of lungo) = the light organ, from Proto-Indo-European leng = not heavy, agile, nimble, whence ultimately also light.
So called perhaps because in a cook pot lungs of a slaughtered animal float, while the heart, liver, etc., do not.
Greek pneumo and Latin pulmo both stem from “to blow/breathe”.