I just did it without downloading any code to my laptop and using only the GitHub site.
The solution looks the same as @swcool’s, but I want to add about the default branch.
In my case, the name of the renaming branch did not exist.
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Change the default branch (to the old branch you want to rename)
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Create a new branch (with a new name)
This action will copy all the contents of the default branch (the branch with the old name) to the new branch (with a new name). At this time, you have two branches with the same code.
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Change the default branch (to the new one with a new name)
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Delete the old branch