"to deprive (a word or phrase) of its meaning," 1900, from weasel (n.); so used because the weasel sucks out the contents of eggs, leaving the shell intact. Both this and weasel-word are first attested in "The Stained-Glass Political Platform," a short story by Stewart Chaplin, first printed in "Century Magazine," June 1900:
"Why, weasel words are words that suck all the life out of the words next to them, just as a weasel sucks an egg and leaves the shell. If you heft the egg afterward it's as light as a feather, and not very filling when you're hungry; but a basketful of them would make quite a show, and would bamboozle the unwary."
They were picked up at once in American political slang. The sense of "extricate oneself (from a difficult place) like a weasel" is first recorded 1925; that of "to evade and equivocate" is from 1956. Related: Weasled; weasling.