Since the port on the tv works with the same hdmi cord, different device, and you've said yourself the mini dvi is suspect...
Trying connecting the laptop via hdmi to another device, or even another port on the tv.
If that doesn't solve the issue, you've pretty much already said it, the mini hdmi to hdmi needs to be replaced. It may be something more esoteric but I'd put money on a simple adapter malfunction before digging deeper.
My PC has a Nvidia 580 GTX with 2 DVI and 1 miniHDMI outputs. It came with a miniHDMI to HDMI adapter. I am using this miniHDMI output to connect to the HDMI 1 port on my plasma. I have resolutions set to 1920x1080 @ 60hz, but the problem is, I get a lot of noisy green pixels on the screen. Does anyone have any ideas what might be causing this?
Also, it's my understanding that connecting through VGA will not support full HD resolution, correct? Otherwise I would just do that.
The HDMI cable is high speed and tested through a Bluray player through the same port, so I can rule out a bad cable. The only point in the connection that is suspect is the miniHDMI to HDMI connector that came with the graphics card.
Any help or advice is greatly appreciated.