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: 
You should see a prompt when you turn on your computer. If the mouse can't be found,

* Check if the mouse is on.
* Check all other computers to see if one of them is paired with the mouse. If that is the case, unpair it.

If none of those work, try a USB mouse.
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.