Going through an AVR is always tough, even if you had a built-in HDMI port; with that old a Mini if you can connect video directly and only route audio to the AVR I think you'll be much better off. The Gefen that geeji helpfully links to has solved the issue for lots of people here but, can you really justify spending another $129 on an intermediate device (that you may or may not ever need again) just to get an old and these days, a largely inadequate, Mac to work in your setup? Better to sell that Mini off (or repurpose it within your home) add its value to that $129 and upgrade to something newer, better designed to function in a home theater context, like a 2010 or 2011 Mini. Why do I suggest that? Because you mentioned EyeTV.
I have an EyeTV hooked up to the Mini. I got this amplified antenna but my reception is still poor-to-nonexistant in my basement room. What are my options for getting a reliable signal to the computer for DVR functionality?
Ditch OTA and get cable? Seriously, if a better antenna in different locations doesn't get you better line of sight to the towers or less interference, that's all you can do. (I'm on the wrong side of my building from the towers, don't have access to my roof, and cannot get reliable OTA no matter what I do.)
The Intel GMA 950 graphics with your old Mac Mini is borderline for 1080p and overcharges the CPU.
It will NOT reliably work on HD programs with 1080p, although it should work on regular digital TV.
I'll rephrase what he said: The Intel GMA950 in your Mini is borderline for 720p, can barely drive an old 720p native HDTV properly and even with your RAM maxxed out cannot run EyeTV properly. It's just not up to the task, wasn't up to it back then, certainly not now. You need at least a 2.2-2.4 C2D with better graphics than the GMA950 to do a decent job deinterlacing 1080i content with EyeTV. Forget watching "live TV," and forget watching sports, at all, even if you set EyeTV to its second best setting.
I have a TOSLINK cable going from the Mini to the receiver, but I feel as though sound quality is poor. Netflix sounds much better from the PS3. I've tried making changes in Audio MIDI setup but I usually end up losing sound altogether.
Which OS version are you running? Is it only Netflix audio that doesn't seem adequate--what about EyeTV? iTunes? a front end like XBMC or Plex? Can you verify you're getting digital audio correctly from those sources, i.e. do the DD 2.0 and DD 5.1/DTS light up when they should given the video source?
You should know that Netflix audio isn't a standard across all platforms and devices,so NF within a browser on a Mac often isn't as good as the same Netflix title via an Apple TV or PS3...