How to rename a directory/folder on GitHub website?
Actually, there is a way to rename a folder using web interface.
See https://github.com/blog/1436-moving-and-renaming-files-on-github
There is no way to do this in the GitHub web application. I believe to only way to do this is in the command line using git mv <old name> <new name>
or by using a Git client(like SourceTree).