I'm a philosophy graduate with a focus on games and biology. I've been approaching meme studies by working on foundational issues of how the new discipline of meme studies is being developed, especially by looking at how different traditions within meme studies have emerged (and how they can be synthesised).
I focus more on qualitative research but I think "hybrid" research methodologies will become far more important. For example, I think laboratories for memetic engineering experiments will could provide much more useful data about how memes work than (say) scraping memepages. I don't have a lot of experience with quantitative research, and would be thrilled to try out all the different research methodologies everybody is using!
I don't have a good enough system for collecting and managing memes, even though I have thousands of memes saved everywhere (including hundreds of memes I made). It's another feature I think we can develop together (see the booru thread). In terms of understanding and following the development of memes, there's no way around participating in the respective communities. Research based on whatever is left behind after a memeculture has died off becomes biased towards the most readily fossilised aspects of the communities' output, which are frequently also the least interesting aspects. Memes certainly require new methodologies; I think the most important aspect is the memetic literacy of the researchers involved. It will be crucial to have memeculturalists conducting research and researchers participating in memecultures.