Hi Andrew!
I am running OS X 10.8.5 too. I googled your problem of not having "Display Audio" as an option and found this...
http://apple.stackexchange.com/questions/89685/activating-the-apple-thunderbold- display-speakers-when-macbook-pro-is-docked
It is not a lot, but there are two suggestions in the small discussion thread:
1. Restart your Mac Mini and THEN see if sound preferences shows "Display Audio" as an option. If so, chose it before it goes away again!
2. If you have any devices connected via AirPlay choose that device in sound output preferences. It may refrresh your list of Output device options and present "Display Audio" again.
Until I came across the "fix" of choosing "Display Audio" instead of "LED Cinema Display" my only consistent option was to restart my Mac Mini, as you have discovered. But, that would only last for a while and I was back to the same old problem of no sound from the Cinema Display's speakers. Another option that sometimes worked was to unplug the Cinema Display from power and disconnect the USB cable carrying sound to the Cinema Display...wait a minute or two...then reconnect everything. If restarting your Mini does not result in returning the sound option of "Display Audio" you might want to try the second option I just mentioned...disconnect power and the USB sound cable to the Cinema Display. Maybe that will get "Display Audio" to appear as an option..
One last thought...and this is a long shot. You could try going into your Application Folder and select Utilities-->Disk Utillity. Once you are into Disk Utility click on the name of your Hard Drive (unless you have changed it, it will be called "MacIntosh HD.") Then click on "Repair Disk Permissions" and let it go through the process. If that doesn't help, you might also want to click on "Verify Disk" to make sure there are no problems with your hard drive. If there are, you will get instructions that might repair the problem.