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.
If the batteries run out in my wireless mouse it never reconnects automatically and I have resorted to reconnecting a wired mouse to click on the bluetooth icon in the menu bar. Then I found that you can activate the bluetooth icon by pressing Control-F8 (the "Move focus to status menus" in the keyboard shortcuts) and using the cursor keys and the enter key to get to the connect menu item for the mouse.