I didn't know how to phrase my title to make it meaningful to anyone, and I can't really explain it now, so I shall use an example:
The opening phrase on the DSGB website is "It's counting, Jim, but not as we know it". This comes from the Star Trek quote "It's life, Jim, but not as we know it". What is it called when someone does this (mainly for comedic effect)?
Isn't it called a trope, when something is recognized widely enough that even with slight alterations, the reader will still recognize the original quote?
Take "Got Milk?" for instance. You can replace milk with just about any noun and people will still recognize it as an allusion to the original.
You could call it making a reference (or allusion) to the original phrase. This is more general, though, and doesn't only apply to situations where comedic effect is intended.
Looked up sources from Peter Taylor's link and in my mind, Snowclones is the correct term for these new meme-clichés or rhetorical tropes (since both these terms fall short; many of the now modified original quotes weren't memes, or at least they don't have to be. They might become memes from snowcloning them, ie. "catch on", but this is after the fact. This also means that you must agree with that they don't have to already be in popular culture, as mentioned by Daniel. Again, they might become part of pop culture. Their use may be rhetorical, but it's not a property of them. Trope doesn't describe this new phenomenon distinctly.)
Snowcloning is the process.
If you're afraid terming them Snowclones is confusing, you will just have to write it out more logically as "modified catch-phrase" or something like that.
It's one form of reference humor, where the speaker counts on the audience having heard a quote, seen a film, or similar.
I think you will frequently hear it referred to as a play on a quote.
An allusion is more general: it need not be funny. "Pop culture references" is what most people would call them -- if what is referred to is part of "pop culture".
I tried to see if people have described such things as "allusive puns". Apparently, some have.
I would refer to this as paraphrasing:
par·a·phrase [par-uh-freyz]
noun
1. a restatement of a text or passage giving the meaning in another form, as for clearness; rewording.
2. the act or process of restating or rewording.
verb (used with object)
3. to render the meaning of in a paraphrase: to paraphrase a technical paper for lay readers.
(http://dictionary.reference.com/browse/paraphrase)
Paraphrasing in the more comical sense used in the question is where a certain sentence structure is re-used in a way that the original source is still recognisable.
For example:
To paraphrase Spock from Star Trek: "It's counting, Jim, but not as we know it".
When certain phrases are paraphrased often in this way, it often gives rise to a meme.
