The built-in solution for re-naming Django projects in PyCharm has worked for me in the past on small projects, although I haven’t tried it on a large project yet. To use it, select the project folder from the file-tree (by default on the left hand side of the screen), then press shift-F6 to bring up the rename dialogue. You can enter in your new name here and preview the changes, which can help prevent surprises. If you’re happy with the preview, click on the “Do Refactor” button at the bottom of the window.
This is also a useful way to rename Django apps.
You can access the same functionality through right-clicking on the folder/item and hovering over the Refactor sub-menu. Or pressing ctrl+shift+a and searching for rename.