The Git project has a trademark policy. You need permission from them if you want to use it in your brand.

This is not likely to receive permission, since this kind of thing is specifically why the trademark began being enforced—this is a service built on top of the GitHub social network. GitHub login has nothing to do with Git. GitHub issues have nothing to do with Git. GitHub's pull request API has (next to) nothing to do with Git. Trying to brand this as "Git Butler" just serves to further conflate GitHub and Git.