I bring my MacBook Pro back and forth between home and work. I have a Magic Mouse in both places. I guess because they are both the same model, I have to re-pair the mouse with the computer every time I go back and forth. They have very similar names so I'd like to rename them something like "Work mouse" and "House mouse". So far I haven't found a way. Is it possible?
With the new update, you may not find the cog wheel, so instead you can go into bluetooth settings and right-click (click with two fingers) the device you wish to rename.
It should come up with a context menu.
Select Rename, which will open a window where you can rename your mouse to anything you like.
I was having the same issue and noticed that the context menu was different when rename would fail:
I was able to get the full menu (with "Disconnect") and successfully rename the device by not using it while attempting to rename (I was using CTRL-Command to bring up the context menu).