Note: There is some contention as to the applicability of this answer. I am directly addressing this portion of the original question:
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, but in which the latter version becomes more well known than the original?
The best term I found was meme:
A meme is an idea, behavior or style that spreads from person to person within a culture. While genes transmit biological information, memes are said to transmit ideas and belief information.
A meme acts as a unit for carrying cultural ideas, symbols or practices, which can be transmitted from one mind to another through writing, speech, gestures, rituals or other imitable phenomena. Supporters of the concept regard memes as cultural analogues to genes in that they self-replicate, mutate and respond to selective pressures.
The key here is the mutation. In the case of your example, there is a rhyming/assonant mutation and Six Degrees of Separation became Six Degrees of Kevin Bacon.
There are plenty of other examples of meme evolution; many of them thrive in chain emails or on Facebook. The one example that really stuck with me, however, was the "25 Random Things About Me" chain that hit in 2009. The reason I remember it well was because of this article in Slate magazine that compared the fad to a virus:
Late last fall, a chain letter titled "16 Random Things About Me" began to chew its way through Facebook. [...] Then something curious happened: It mutated. Since everyone who participates is supposed to paste the original instructions into her own note, it's easy to tinker with the rules. Soon enough, 16 things (and 16 tagged friends) morphed into 15—and 17 and 22 and 35 and even 100. As the structure crumbled, more users toyed with the boundaries. Like any disease, "Random Things" was mutating in hopes of finding a strain that uniquely suited its host.
Studying the idea of Six Degrees of Kevin Bacon outlasting its predecessor would fit the scope of measuring popularity and catchiness. Much like pictures of Keanu Reeves on a bench or Xzibit's goofy phraseology, concepts morph until they stick in the popular consciousness. Naturally, rhyming makes a logical next step in the evolution of a phrase. So would swapping out a generic word (separation) for a specific cultural, er, icon (Kevin Bacon).
Of note, Wikipedia even calls the it a meme:
Though [Bacon] was initially dismayed by the game, the meme stuck, and Bacon eventually embraced it.
While a more specific term may be found, meme certainly works as a backup umbrella term.
A more specific alternative: The Weird Al Effect
When a parody remains popular after the original works being parodied are no longer known to the audience.
Named for the fact that, when listening to the earlier work of "Weird Al" Yankovic, modern fans may be so unfamiliar with the songs being mocked as to not even realize that the Weird Al song is a parody. For example, many people are now more familiar with "I Lost on Jeopardy!" than with the original "Jeopardy" by the Greg Kihn Band.
This is, again, not a specific match based on rhyming but the idea of a variation becoming more popular than its original again matches. The effect also carries a heavy implication that the child term survives the parent term's fade into obscurity.
Best Answer
Consider the following: