[We like add answers to some anonymous support questions to our blog, when we feel they might be helpful to others.]
"I am trying to connect my Mac Mini to my 1080i HDTV. The results are terrible with poor resolution and huge overscan. Any ideas?"
OK, this could be one of several things. I think the TV is unhappy with the resolution being sent by the Mac Mini.
First of all the Mac mini has come with a variety of video graphics "chipsets" from ATI. Nvidia and Intel. Which chipset you have determines which HDTV friendly resolutions are available. If your has the Intel embedded graphics chipset, you may have no good choices.
Second, you mention your TV is 1080i compatible, but don't mention 720p. Is it compatible with 720p?
Displaying computer images is always best with progressive scan mode, the "p" in 1080p, rather than interlaced, the "i" in 1080i.
So, if the TV is OK with a 720p signal, you should try setting the Mac Mini's resolution to 1280x720@60Hz. That should be acceptable to a 720p capable TV.
If the TV only likes 1080i (and lower resolutions like 480i/480p) then you should try seeing if there is an updated driver you can download for your Mini's graphics chipset. ATI and Nvidia probably have decent 1080i resolution options with overscan control settings. If you still can't get things perfect there's "switchresx" which can sometimes help.
If your Mac Mini has the Intel graphics chip, then you may be stuck with using 480p with your current TV.