Much has been said about the remix, but riffing on ideas - specifically internet memes, is a slightly different beast. An original idea that resonates might just inspire someone to put a spin on it, extending or enhancing the idea. This is perhaps more common than is generally recognized, and I would argue, a growing trend.
For example, I was recently directed to a clever image that poked fun at Twitter culture on a day that Twitter was suffering performance issues. This image resonated with me because I TOO was affected by these issues and was inspired to attach my own meaning and create a different image that poked fun. This, in turn, inspired a friend to create more, clever interpretations of the idea...
(from Mykl Roventine)
(From Keith Hopper)
(from Andy Carvin)
This is only one example of an expressive idea train, where each of us saw different meaning and chose to share that meaning in a slightly different way. Based on how a specific idea might inspire you (and towards what ends), modification and republishing of a meme might manifest as a remix, knockoff, spinout, or analog of the original idea, described as follows:
Remix: Taking a single idea and modifying the orginal content. For example, you might take a funny image and give it a soundtrack, or mash it up with a video, making it funny in a new context.
Knockoff: Same idea, different name. Generally done by someone who perhaps wants to suggest they originated it.
Spinout: Different idea, but with a common source of inspiration, such as a topic - like different jokes based on the same high-profile cultural event.