You don't make it clear what kind of TV service you have. As you appear to be American I can deduce that your TV service could be one of the following -
1. ATSC - a digital over-the-air service. Being digital all you need is the EyeTV 250 Plus, or the EyeTV Hybrid (USA), or you could get the HDHomeRun see
http://www.silicondust.com/products/hdhomerun_atsc
2. Clear QAM cable. The same tuners as above should work.
3. Digital Cable Card. Currently there is no Mac solution for this and unlikely to be so for the foreseeable future. You only option here is to use the Cable providers set-top box and connect a Mac via an analogue tuner. The EyeTV 250 Plus would do this job and would off-load the work of digitising the analogue output.
4. DirecTV or Dish TV (both are satellite systems). Here in Europe (and most of the rest of the world) all the satellite providers use the same basic standard of DVB-S and/or DVB-S2 and there are several suitable tuners available, however as far as I am aware, DirecTV and Dish use their own weird variations and as the US satellite TV market is minute compared to the global DVB-S/DVB-S2 market no suitable tuners are available. As such again your only choice is to use an EyeTV 250 Plus to digitise the analogue output of a DirecTV or Dish TV set-top box.
Note: having to do an analogue capture from a set-top box is so 20th Century, furthermore it will never offer HD quality.
If you are using a digital connection, then your old Mac mini will have no problems coping, and even if you use an analogue connection, as long as you use the EyeTV 250 Plus the work load is off-loaded to the tuner and again your Mac will be able to cope. If you use used analogue, and an old tuner like the EyeTV 250 (not plus), or the EyeTV Hybrid then your Mac would potentially struggle.
I have only ever used digital TV feeds and even an old PowerMac G4 has no problems.