Looking at the GitHub API reference at http://develop.github.com/p/repo.html, I see all sorts of good stuff, but I don\'t see any way to rename a repository through the API.
Create some variables for clarity:
user=MyUserName pass=MyPassword newName='{"name": "NewNameForRepo"}' oldName="MyRepo"
Then use curl to make the request:
curl -u "$user:$pass" -X PATCH -d "$newName" https://api.github.com/repos/$user/$oldName