Let me break down your questions

(1) Is there a term that succinctly represents the transformation of a word or phrase through assonance or rhyme into another word or phrase of similar meaning?

 

(2) As (1), but in which the latter version becomes more well known than the original?

 

(3) Failing that, perhaps this falls under a larger category of word or phrase substitution.

For (3) - it sure does, for example, in classical rhetoric the substitution as one of the four categories of change indeed recognizes various figures

• it could be thought of as periphrasis or antonomasia; though in essence periphrasis is introduction of a name to take properties associated with person or thing behind the name and apply them to the subject, where in the given case it is only partially so

There is another explanation, the mechanism at hand is actually not substitution, but subtraction, more specifically omission, actually ellipsis from

Six degrees (of separation) of Kevin Bacon

or in another example, "Six degrees (of separation) of Obama." which seem the same in the actual meaning, but with "omission of a word or short phrase easily understood in context." - which is the exact definition of rhetorical ellipsis.

Footnote:
_{As for answer to (1) I believe that you are not precise enough (you say assonance or rhyme, but I believe that you actually include many other types of wordplay here, isn't it so?). If I am right in my assumption then call it wordplay; if I am not right please refine further what you mean.}

_{Similarly unsatisfactory is my solution for (2): the popularity of a term is very hard to measure. You could do it for the terms with largely overlapping meanings (which is not the case here) by measuring frequency in certain corpus, or if a specific terms fall into categories of 'general knowledge' or 'general culture' while the other term does not, but I am not aware of any specific words other than: popularized/rare (and so on). Or maybe successor if you want to emphasize that the words came later (and it does imply that it overtook the phrase). But, these are trivial.}

Let me break down your questions

(1) Is there a term that succinctly represents the transformation of a word or phrase through assonance or rhyme into another word or phrase of similar meaning?

(2) As (1), but in which the latter version becomes more well known than the original?

(3) Failing that, perhaps this falls under a larger category of word or phrase substitution.

For (3) - it sure does, for example, in classical rhetoric the substitution as one of the four categories of change indeed recognizes various figures

• it could be thought of as periphrasis or antonomasia; though in essence periphrasis is introduction of a name to take properties associated with person or thing behind the name and apply them to the subject, where in the given case it is only partially so

There is another explanation, the mechanism at hand is actually not substitution, but subtraction, more specifically omission, actually ellipsis from

Six degrees (of separation) of Kevin Bacon

or in another example, "Six degrees (of separation) of Obama." which seem the same in the actual meaning, but with "omission of a word or short phrase easily understood in context." - which is the exact definition of rhetorical ellipsis.

Footnote:
_{As for answer to (1) I believe that you are not precise enough (you say assonance or rhyme, but I believe that you actually include many other types of wordplay here, isn't it so?). If I am right in my assumption then call it wordplay; if I am not right please refine further what you mean.}

_{Similarly unsatisfactory is my solution for (2): the popularity of a term is very hard to measure. You could do it for the terms with largely overlapping meanings (which is not the case here) by measuring frequency in certain corpus, or if a specific terms fall into categories of 'general knowledge' or 'general culture' while the other term does not, but I am not aware of any specific words other than: popularized/rare (and so on). Or maybe successor if you want to emphasize that the words came later (and it does imply that it overtook the phrase). But, these are trivial.}

Let me break down your questions

(1) Is there a term that succinctly represents the transformation of a word or phrase through assonance or rhyme into another word or phrase of similar meaning?

(2) As (1), but in which the latter version becomes more well known than the original?

(3) Failing that, perhaps this falls under a larger category of word or phrase substitution.

For (3) - it sure does, for example, in classical rhetoric the substitution as one of the four categories of change indeed recognizes various figures

• it could be thought of as periphrasis or anatomasia; though in essence periphrasis is introduction of a name to take properties associated with person or thing behind the name and apply them to the subject, where in the given case it is only partially so

There is another explanation, the mechanism at hand is actually not substitution, but subtraction, more specifically omission, actually ellipsis from

Six degrees (of separation) of Kevin Bacon

or in another example, "Six degrees (of separation) of Obama." which seem the same in the actual meaning, but with "omission of a word or short phrase easily understood in context." - which is the exact definition of rhetorical ellipsis.

Footnote:
_{As for answer to (1) I believe that you are not precise enough (you say assonance or rhyme, but I believe that you actually include many other types of wordplay here, isn't it so?). If I am right in my assumption then call it wordplay; if I am not right please refine further what you mean.}

_{Similarly unsatisfactory is my solution for (2): the popularity of a term is very hard to measure. You could do it for the terms with largely overlapping meanings (which is not the case here) by measuring frequency in certain corpus, or if a specific terms fall into categories of 'general knowledge' or 'general culture' while the other term does not, but I am not aware of any specific words other than: popularized/rare (and so on). Or maybe successor if you want to emphasize that the words came later (and it does imply that it overtook the phrase). But, these are trivial.}

Let me break down your questions

(1) Is there a term that succinctly represents the transformation of a word or phrase through assonance or rhyme into another word or phrase of similar meaning?

(2) As (1), but in which the latter version becomes more well known than the original?

(3) Failing that, perhaps this falls under a larger category of word or phrase substitution.

For (3) - it sure does, for example, in classical rhetoric the substitution as one of the four categories of change indeed recognizes various figures

• it could be thought of as periphrasis or antonomasia; though in essence periphrasis is introduction of a name to take properties associated with person or thing behind the name and apply them to the subject, where in the given case it is only partially so

There is another explanation, the mechanism at hand is actually not substitution, but subtraction, more specifically omission, actually ellipsis from

Six degrees (of separation) of Kevin Bacon

or in another example, "Six degrees (of separation) of Obama." which seem the same in the actual meaning, but with "omission of a word or short phrase easily understood in context." - which is the exact definition of rhetorical ellipsis.

Footnote:
_{As for answer to (1) I believe that you are not precise enough (you say assonance or rhyme, but I believe that you actually include many other types of wordplay here, isn't it so?). If I am right in my assumption then call it wordplay; if I am not right please refine further what you mean.}

_{Similarly unsatisfactory is my solution for (2): the popularity of a term is very hard to measure. You could do it for the terms with largely overlapping meanings (which is not the case here) by measuring frequency in certain corpus, or if a specific terms fall into categories of 'general knowledge' or 'general culture' while the other term does not, but I am not aware of any specific words other than: popularized/rare (and so on). Or maybe successor if you want to emphasize that the words came later (and it does imply that it overtook the phrase). But, these are trivial.}