Answer
1. If ArcMap or ArcCatalog is open, close the program (the add-in remains temporarily cached until the software is closed).
2. Make the necessary changes to the Python script in the installation folder of the add-in.
3. Run the makeaddin.py script in the add-in folder to recreate the .esriaddin file.
4. Double click the .esriaddin file to reinstall the add-in.
5. Restart ArcMap/ArcCatalog.