If you have multiple instances of a prefab in a scene, delete the root prefab from your assets folder, then re-link the objects in the scene by creating a new root prefab, some of the objects in the scene will have their name changed whenever you change the name of the root prefab. It looks like something goes wrong when a new base prefab is created and the objects in the scene are (presumably) re-instantiated.

1. Install Unity
2. Open the attached project
3. Open "Scene" Scene
4. Drag multiple instances of the "PrefabObject" prefab into the scene
5. Delete "PrefabObject" from the assets folder
6. Drag "PrefabObject" from the hierarchy to the Project window to create a new prefab. Note that at this point all the prefab objects in the scene are renamed and presumably re-instantiated
7. Open the "PrefabObject" in Prefab Mode
8. Rename the root object and save it
9. Return to the normal Scene view
10. Objects in the scene have been renamed to the name chosen in step 8