Renaming files in PyCharm is simple. You simply select Refactor > Rename when right-clicking on a file in the tree.
This will open a popup where you can type in the new filename. There are additional options when renaming, such as searching for references and in comments, strings, etc.
NOTE: While PyCharm is indexing files, the option is unavailable. Once indexing is finished (can take a while), it becomes available again (thanks @Eric_Sven_Puudist).