Old Apple Magic Mouse is discoverable, but not pairing

I have a couple of old battery-powered Magic Mice, which both sometimes suffer from the same problem: they get disconnected, then when I attempt to re-pair the affected mouse, the Mac tells me: “Couldn’t connect to the device”.

The first thing to try is to hold the mouse button down while trying to pair (with the mouse ON of course).

If that doesn’t work, shut off Wifi, shut off BlueTooth, then turn bluetooth back on and hold the mouse button down, then click “Pair”. Don’t let up on the mouse button until it is paired. Then you can turn Wifi back on.