We like TortoiseGit, and want to take advantage of GitHub's 2-Factor Authentication...but the two don't seem to want to work together.
The GitHub blog post about 2-Factor Authentication mentions that you can generate a "personal access token", that can be used in place of a password when 2-Factor Auth is enabled:
I'm not sure if it will actually work with TortoiseGit, but maybe you can give it a try? You can generate an access token at the Authorized applications settings page: