Note: I am *not* recommending the solution below, I'm just putting it out there for context.
I have a 2008 Core 2 Duo notebook with crappy x4500 chipset graphics. However, the notebook does have HDMI output. I'm a big Linux guy, so I have gotten 1080P video output from the notebook to my HD-TV on a KDE 4 desktop including audio channeled over the HDMI, and the video plays just fine (I recommend VLC or a fancy media-centric Linux distro like XBMC).
Now obviously video playback is not the only thing that people do with HTPCs, but you really don't need to have a huge CPU or GPU to do the basics. If you want to really use this HTPC for games, then you need to look into better GPU solutions, but practically any modern system can do the basic stuff just fine. The fancier points come into price/performance ratios and having a quiet system.
4770K @ 4.7 GHz; 32GB DDR3-2133; GTX-1080; 512GB 840 Pro (2x); Fractal Define XL-R2; NZXT Kraken-X60
--Many thanks to the TR Forum for advice in getting it built.