You have a project in Visual Studio and you want to publish to GitHub for example. From Team Explorer Home you click Sync which takes you to the screen below.
You click Publish to GitHub enter the target repository name and description and click Publish.
Once completed you go back to Team Explorer Home.
You realised that the repository name needs changing for one reason or another. Click on Settings.
Click on Repository Settings and scroll down to Remotes.
Click Edit and change the URL.
Note: The new repository does NOT exist so any pushes will fail at this point. Head over to GitHub and open the original repository.
Click Settings and rename the repository to the new name. Now all pushes from Visual Studio will succeed.