A mondegreen is rarely a pun, but an unwitting aural mistake, a distortion that produces a distinct change in meaning. The humor mostly arises from our recognition that the words and meaning aren't correct, rather than because the words have a clever double meaning or lack sense - though the latter case (when sincere) adds to our derisive delight. A listener must usually have prior knowledge of the song to catch a mondegreen.

Consider the case of Lady Mondegreen herself: this distorted line makes perfect sense, except that her cameo death is not preceded by her appearance anywhere earlier in the song, in actual events or, indeed, in history.

Anyway, to construct a convincing mondegreen (oxymoronic though that is), overt punning should be avoided. To be most authentic, you could play a song you aren't familiar with through bad speakers at low volume, and write down what you hear, striving to make moderate sense of the iffy parts. Or you could sing a song to a young kid, having him repeat after you line by line. Not quite according to this challenge as posed, but it would probably produce a couple howlers. (Just don't howl at the kid; you don't want him to develop a complex.)