Rename the local branch to the new name
git branch -m <old_name> <new_name>
Delete the old branch on remote - where is, for example, origin
git push --delete <old_name>
Or shorter way to delete remote branch [:]
git push :<old_name>
Push the new branch to remote
git push <new_name>
Reset the upstream branch for the new_name local branch
git push -u <new_name>