A term describing the replacement of a specific word in a saying with one that ryhmes
Solution 1:
The term that describes this phenomenon is snowclone, which Collins defines as
a verbal formula that is adapted for reuse by changing only a few words so that the allusion to the original phrase remains clear
Note, however, that this definition says nothing about the extent to which the new word(s) may or may not rhyme with what they replace.
According to Collins, snowclone is a 21st-century coinage which derives from the common formula "If the Inuit have N words for snow, then surely X have Y words for Z" + clone.
Solution 2:
The Jargon File describes this as "soundalike slang".
Hackers will often make rhymes or puns in order to convert an ordinary word or phrase into something more interesting... Terms of this kind that have been in fairly wide use include names for newspapers:... Wall Street Journal → Wall Street Urinal
Arguably, soundalike slang refers to accepted slang terms for things, woven into otherwise normal speech, whilst snowcloning refers to a specific joke.
Solution 3:
From http://dictionary.reference.com/browse/pun we have:
the humorous use of a word or phrase so as to emphasize or suggest its different meanings or applications, or the use of words that are alike or nearly alike in sound but different in meaning; a play on words.
This is in accord with what I've always called what you describe: a pun. Substituting a word for a rhyme counts as using a word that sounds alike. It's almost the most basic form of pun there is. We might say 'the right to arm bears' (the same words have different meaning) or 'the right to bear farms' and have a picture of someone uncaging a bunch of bears in a farmer's frontyard (now a similar sounding word has different meaning, too).
So, you can safely call this a pun and be understood. I appreciate the more recent evolution of a more specific term of 'snowclone', as well, but I'm not sure it quite covers the sense of lyricalness. However, I'm pretty sure the wit is inherent in the choice of using an allusion.
I think soundalike slang is more appropriate when the usage is more about defining the language of your group and less about wit--when it becomes a formula.