If your remote repository is on GitHub, you can rename the branch over the web interface: Go to the branches list for the repository and click on the pencil symbol on the right.
If you don’t want to do that or your repository is hosted somewhere else and that place doesn’t offer renaming, you can push to a specific remote branch name, with or without renaming your local branch. I don’t know if GitHub Desktop supports this, but
git push lets you specify local and remote ref names separately (though the default is to use the same for both). For example,
git push origin HEAD:my-branch
pushes the current HEAD to the remote branch
my-branch. The documentation has a detailed description of the
<refspec> format, and examples further down.