Mac Mini HDMI port?

No the HDMI port only supports output.

If your Mini has Thunderbolt ports you can purchase one of thes

http://eshop.macsales.com/item/Blackmagic%20Design/BINTSTBEXT/

You can also get video input to the Mini using an USB device but I suspect that the an USB device will not support high resolution

Due to HDCP content protection which is used on most HDMI systems, it is unlikely the BlackMagic Intensity card will work as desired to input video from a TV set-top box, it generally would work with Games Consoles and Cam-Corders or similar devices.

There is the SiliconDust HDHomeRun PRIME which supports encrypted CableCard services, see - https://www.silicondust.com/products/hdhomerun/prime/ however it is not clear how well if at all this works on a Mac. I mention it because at least some of their products do work on a Mac including an un-encrypted CableTV tuner, see https://www.silicondust.com/products/hdhomerun/hdhomerun-connect/

Note: The HDHomeRun devices effectively act as a setup-box i.e. tuner.

The reason the PRIME model may not work is ugh yet again content protection. Ironically you may find it more likely to work with an iPhone or iPad!

There is also the SlingBox 500, see - http://www.slingbox.com/Products/SlingTV/Features.aspx this might do what you want by 'copying' and converting the output of your existing set-top box and sending it to your Mac to be displayed, it does not replace your existing set-top box. Again it is not clear how well this will work with HDMI sources due to the HDCP content protection, it would work fine with component connections but these days most set-top boxes no longer come with component connections deliberately to prevent you doing this. Again scummy anti legal paying customer measures gone mad.