In order to check the status of the Applecare warranty, I needed the serial number. This system had been previously removed from its stand and connected to a mounting arm, so I couldn't guarantee the hardware matched what was printed on the bottom of the stand. Uh-oh.
Fear not! The Mac actually has a number of fairly awesome tools available if you know where to look. The system I had connected to it was running 10.9 and the system information application wouldn't tell me the serial number of the display.
The workaround is to open the terminal, then run:
sudo system_profiler SPDisplaysDataType
This spits out some helpful information, and buried in that information dump is a section called Displays. In that is "Thunderbolt Display:", and under that is the serial number!
Seriously handy for system information, especially if you have to ssh into a user system and look up hardware information. For more information take a gander at: