What monitor do you have? If it is an Apple monitor, then see if you can get the exact model by entering the monitor's serial number on the check coverage page here:
Check Your Service and Support Coverage - Apple Support
Do you get any LED power when the display is powered on? It should be on solid, but it if the display's power LED is flashing, then it means the display has a hardware issue. Many Apple displays have the display cable hardwired so you cannot easily swap cables.
FYI, that is a DVI connector.....both on the display cable and the back of the Mac Mini.