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.
I'm a humanities grad, and right now my fields are computation arts, sociology and communications. I have a background in nonfiction writing and media/game studies. What I'm trying to figure out how to do is to bridge the qualitative explorations of memes with actual quantifiable data--yet still base that data collection/processing/analysis on a qualitative theoretical analysis. So for example, one thing I've been looking at is the use of sad cat and pepe memes on Reddit by scraping with python. After retrieving the data, I look for trends, most commonly used words, and sentiment. Then I analyze the most popular memes.
I'm really taken with Stuart Hall's theory of articulation and the idea of memes as holding/maintaining linkages that are useful, but not essential. Something strikes me as interesting in terms of how memes function as information passed through platforms and vectoral capitalism. I'm also thinking of how to use Digital Methods (Richard Rogers) to form a more comprehensive understanding of memes and their effects on social and cultural change.
But I guess the biggest thing that I'm considering as I try to construct a methodology for my diss and meme projects is how to theorize memes to get an exciting, epistemologically sound set of ideas to collect data around to prove/disprove/explore memes as they are rather than memes as "meme magic"
I'm still working on developing my reddit scraper and will share the github repo once it's more complete. It's a one trick pony, but a useful one trick pony.
For quantative methods, I strongly recommend Knobel, M., & Lankshear, C. (2018). Online memes, affinities and cultural production (2018 update to our 2007 chapter) To appear as: Knobel, M. and Lankshear, C. (forthcoming). Memes online, afinidades e produção cultural (2007 – 2018). In Chagas, Viktor (ed.). Estudos sobre Memes: história, política e novas experiências de letramento. 2019.https://doi.org/10.13140/RG.2.2.34717.77280
It's a bit of a mix, but quite feasible in general.
For digital methods, Marc Tuters might be a good reference for checking out. Marc has written quite some stuff about memes, 4chan and post-truth. A bit context-specific, but interesting. https://journals.sagepub.com/doi/full/10.1177/1461444819888746