What version Mac Mini would I need...
Staying in OS X, anything you have can handle iTunes music, even a single G4. If you want optical out to your AVR that means at least an Intel Mac Mini or Macbook. For recording and
playing back OTA HD with the most common OS X recording software EyeTV, you'd need a fairly decent Core 2 Duo, no older than the 2.53 2009 Mini. Any Mini from 2010 to 2012 would do a great job for you in the home theater. An older C2D Mac can record, that doesn't take much effort and do a passable job playing back 1080i recordings, but if you're at all a critical viewer you'll see flaws with anything in the 1.83 to 2.2 C2D range.
An HD Homerun is a dual tuner that can do ATSC which EyeTV supports:
so if you're content recording two different channels simultaneously, go with that. You may have to buy the software separately from El Gato, because I don't think El Gato sells the HD Homerun directly anymore. You can load up multiple single tuners, sometimes I have three EyeTV 500s daisychained over firewire to the same Mac, but it can be a bit of a dark art figuring out which tuners work well together. It isn't actually supported by El Gato. If I were starting over, I'd go with the dual HD Homerun.
...plays British BBC shows recorded from online BBC iPlayer on the TV.
I, too, watch the BBC iPlayer (and ITV and Channel 4) via a Mac in the living room; however, I don't know a way to "record" those streams. EyeTV won't record Safari internet streams.