Allowing the user to supply the repository endpoint would also immediately allow people to use a GitHub for Enterprise installation (via the same API)...

General git(svn/hg...) connectivity would no doubt be useful for some, but I think would be complex overkill for most; GitHub integration, however, would be fantastically useful for OSS developers, or indeed anyone using GitHub rather than DropBox to keep documents under version control in the cloud. At the very least, GitHub support would be a great (and far simpler) first step.

Integration via the GitHub ReSTful API requires no more than JSON over HTTP(S), exactly like DropBox (which you already support), and hence no extra client-side libraries on iOS, although as of last week GitHub have indeed extracted a free/MIT Objective-C library from their own Mac client. You could beat them (and no doubt, soon others) to an iOS client!

Links:

GitHub via HTTP

DropBox ReSTful API

GitHub API via Obj-C