If it does work with the AV/RCA/Phono lead and you can get QuickSupport up and running (open Google's Play store, sign in, search for and install Quick Support, open the app, install any add-ons asked for), then install Team Viewer on your PC and connect to your DroidBOX.
Head to the Screen/Display settings area, and disconnect the AV lead, now connect HDMI. On your PC, look for any reference to auto selecting HDMI mode, and try turning it off if present. Now change the resolution used to 720p@60 or 50 Hz, then 1080 at 60 and 50Hz. Hopefully one of the HDMI modes will work OK on your TV.