Apple Magic Mouse is discoverable, but not pairing

Just got Apple Magic Mouse from eBay. I cannot figure how to pair it with my Mac Mini: it's discovering fine, but when I click on Pair button, my Mac is thinking for a little while and says: "Couldn't connect to the device".

When I'm trying to pair the mouse with Windows 7, I get a different error: Authentication failed. The code I entered is default 0000.

The mouse was described as New (other), but it seems not. It already has a name: Cora's mouse.

Is there any way how can I get it working?

My issue was that my Magic Mouse was discoverable through the Bluetooth Preferences pane, but when I clicked 'Pair', it couldn't connect. After an hour, I found a solution that worked:

1. Turn off the Magic Mouse
2. Open the Bluetooth Preferences
3. Delete your mouse's profile
4. Be sure the Bluetooth Prefs pane is searching (indicated by the spinner)
5. Hold the mouse button down, then flip the switch to turn it on (it should blink green during this time
6. While still holding down the mouse button, click 'Pair' in the Bluetooth Prefs pane

Follow these instructions:

1. Turn off the mouse by holding down the power button until the light goes out.
2. Turn on the mouse by pressing the power button, and keep the power button pressed until it is paired. The light on the mouse will start blinking, it will show up in Bluetooth Preferences, and you will be able to pair it almost immediately so long as you do not let go of the power button from the moment of turning it on until after the mouse is paired.

I would change the batteries in the mouse and see if that fixes anything.