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).
You can!Just press edit as per @committedandroider's original post and then hit backspace with your cursor at the start of the filename. It will let you then edit the folder. When done hit forward slash to then edit the filename again.
