Disconnect the coax cable and ethernet (if connected). Make sure the HDMI cable is firmly connected on both ends (it can back itself out). Pull the power for like 10 seconds, then plug it back in for a fresh start. Often by doing this you can get by whatever the hangup is... once booted, plug the cable(s) back in.
Beyond that, I would try another HDMI cable, and borrow the power adapter from one of your other minis just to eliminate all possibilities and then finally try connecting the Mini to another tv... The hdmi port on your tv might be starting to go south and Tivos have been showing some sensitivity to HDMI problems. If that testing gives you vague or confusing results, you also could cross-test by putting another Mini in this one's spot and see how it behaves.
Beyond that, there might be some tricks to get into boot-time tests that I am not fluent in... but it might be time to declare it toast at that point.
-Roamio Pro/Slide Pro
-Stream (Gotta go unplug the sucker one of these days!)