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.
In addition to other answers that indicate to use a secondary click on the device shown in the Bluetooth menu, also note that this is not possible while the device is connected with the cable.
I obtained a new Magic Mouse and connected it using the cable to my computer running macOS 10.14.3. The device was recognized and a system dialog informed me that "$PreviousOwner's mouse" had been connected to my system over Bluetooth and was ready to use. Naturally I wished to rename it, but the contextual menu only contained the option "Remove". There was neither "Rename" nor "Disconnect".
After having removed the cable and reopened the Bluetooth menu, the contextual menu showed the two options that were previously missing, and I was able to proceed with the rename without further problems.
I later reconnected the cable and noted that the two options again disappeared.