Question: My bluetooth mouse batteries died, and when I replaced them, the mouse didn't reconnect. Now how can I get OS X to discover and connect my mouse without having a mouse to open the mouse dialogue?

Answer: 
Make sure your mouse has been reset.

1. Press `⌘`+`Space` and type "Bluetooth" to open bluetooth in System Preferences. You will see your mouse in the list of devices and a button to 'Connect'.
2. Use the `Tab` key to move to the next control, this will highlight the next one for you (If you are stuck on the text field, press `Ctrl`+`Fn`+`F7` to switch between navigation of all controls on the screen, press `Ctrl`+`F7` to switch back).
3. Once nothing is highlighted in the window, press the `Space` bar, this in turn selects the 'Connect' button for you.
Keep a cheap usb mouse and keyboard in the back of some closet (with the batteries removed) to quickly use in an emergency. I've successfully used the keyboard and mouse once each in the last 8 - 10 years but it's worth it.