Johnfoster, my understanding is that while the two Mac Mini models (i5 vs i7) both use the same graphics card the specs I see posted give their respective peformance in graphics different... not by alot but some.
Huskermn, yes the EyeTV is HD capable up to full 1080p.
I've been researching this topic and from what I can tell, for 2D graphics (video play of movies, even HD) both Mac Mini models are supposedly very adequate. This in particular I'd like to get confirmation from real world tests or reports by users who have both the 2012 Mac Mini and EyeTV playing 1080p HD content.
Where they say the Mini chokes somewhat is on 3D graphics support for video games that are trying to render frames of 3D content. Since I don't play those games nor would I ever on this machine dedicated to the HDTV, I'm not bothered by that.
Some have suggested I go with a Windows hardware solution but I'm an all-Mac household and like to keep things tightly integrated and networked without issues with compatibility.
If anyone can comment on any of this let me know.