For starters, this problem is new; the setup I now have (MBP, this TV, this cable) used to work fine in the past.
I plugged my HDMI cable into the TV (Sony Bravia), then into my mid-2009 13" MBP. It was already set to mirror displays, and the TV did just that, and everything was fine. BUT: after a few minutes, the TV screen lost my Mac's display, and went black, except for the text "HDMI2" (which was correct). My Mac correctly registers that this display is connected ("Sony TV", with the correct resolution), seems to react as it usually does when I reset mirroring, and otherwise seems OK. On the other end, the TV also otherwise works properly as a TV (good reception and picture). Unfortunately I have no other Mini Display Port-to-monitor/projector cables around, not do I have any other monitor with a HDMI socket here, so I can't check whether that would work.
One additional point, which I'm unfortunately unsure of: I think I heard a soft, almost-inaudible click at the moment the TV pic disappeared, and I believe (if it was there at all) it came rather from the Mac than from the TV.
Repair attempts: I've tried to re-connect the two in all possible sequences - Mac and TV both on, then connect; Mac on first, connect, turn TV on; etc. - all to no avail. Run through all possible settings on Mac and TV. Even rebuilt my directory with DW (you never know!).
Anyone have any idea what might be going wrong, or which part - Mac, TV, cable - might be at fault?