For when you don’t want to fork; you need to change code ownership.
git clone git@github.com:originaluser/originalrepo.git
cd originalrepo
git remote rm origin
git remote add origin git@github.com:newuser/newrepo.git
git push origin master
That was easy. Keep in mind this is just the repo. AFAIK there is no way to migrate the wiki or the tracker.
For all your devs who need to switch over to the new repo it should just be a matter of:
git remote rm origin
git remote add origin git@github.com:newuser/newrepo.git
Remember to take care of your branches as well. I wonder how you do this with hg and bitbucket.