Here's what we know: Rename of actions does not "stick".
Here's what we don't know:
1. Photoshop version?
2. Operating system (Mac or PC)?
3. When does the action reverts to old name - immediately?, after close and re-launch?, randomly?
4. How the rename is done - looks like the standard double-click in the Actions pallet, but not sure.
5. What are the before and after action names? Any strange characters being used?
So, a wild guess:
In Windows CC versions, Photoshop writes the entire actions set to a file called "Actions Pallette.psp" in the Adobe folder tree somewhere under "C:Users\AppData\Roaming".
It was somewhere else in older Photoshop versions (can't remember where). Don't know where it is on Mac. But more important, it writes/updates this file ONLY when Photoshop exits. If something goes wrong during the termination of Photoshop, the file will not get updated. Any changes to actions made during the session will appear to revert when Photoshop is re-launched.
So for the OP, find this file. Check it's time stamp. Open Photoshop, rename an action, close Photoshop. Re-check the file time stamp. Did it change? Did the action name revert anyway?