First checkout to your old branch:
git checkout old_branch
To be sure that you have a branch up to date:
git pull origin old_branch
Rename local branch:
git branch -m old_branch new_branch
Set upstream to track your new named branch:
git push -u origin new_branch
Remove old remote branch:
git push origin :old_branch
-m option is clearly explained in GIT documentation:
-Moption, <oldbranch> will be renamed to <newbranch>. If <oldbranch> had a corresponding reflog, it is renamed to match <newbranch>, and a reflog entry is created to remember the branch renaming. If <newbranch> exists, -M must be used to force the rename to happen.
You can also use
-c option to copy your branch instead of renaming.