As Peter said, if what you are making are groups, you can edit one without affecting other copies no matter what the name is. You won't get an option for Make Unique if the thing you have selected is a group. That's a component thing only.
I just made a test file and had no problem changing the name of a copy of a nested group to something else. Are you trying to type the new name in Outliner or in Entity Info? You can't do it in Outliner so if that's where you are trying to do it, it's no wonder it doesn't work.
If that's not it, you'll have to provide more information. The SKP file, screenshots, something.