Here are my two cents on using GitHub in the classroom.
There is the obvious use for assignments that require students to write code. In fact, that seems the default use envisioned by GitHub,
However, in my own work, I have been using GitHub to keep track of writing projects, such as latex-based papers. Based on my experience, I think the system can also be used by students for (text) writing assignments. For example, students could use any text-based writing system (txt, tex, md) and push/commit their work in progress. This would allow them to save snapshots (no more files sunday_morning_12.docx). I could check in on their progress. Branches could be used for different approaches or flow of argumentation. A more elaborate use would be to use git as a kind of “track changes” feature in Word.
My intent is to test-drive this idea with some grad students and see whether this works for them. If so, I might roll this out to undergraduate students.