For TIVO you want
* lots of hard drive space, preferably quiet drives like Samsung Spinpoints and Seagates. PATA/IDE vs. SATA makes no difference. DVD-quality recording takes about 2 GB of space per hour, for programs you care less about you can use 1 GB / hour recording to save space.
* TV card with hardware MPEG2 encoding and preferably onboard audio capture (instead of using soundcard). The Happauge PVR- series is good, there is supposed to be a new PVR-150mce that's under $100 and designed for media center. Cards using software encoding like the $40 Leadtek tend to have dropped frames and audio sync problems.
* video card with good TV-out.
* DVD burner if you want to archive programs, or to capture and burn old VHS or analog camcorder tapes (note that commercial VHS tapes are copy-protected).
* quiet CPU heatsink like the Zalman 7000 alcu
* quiet power supply from Zalman, Antec, etc.
For a quiet power supply, you either want micro-ATX and a case that can accept standard power supplies (Cooler Master makes some nice mATX desktop cases) or a standard mid-tower case, preferably using 120mm case fans like the Antec Sonata.
With hardware MPEG2 encoding, raw CPU is much less important, so you should be thinking of one that is reasonable (2000+) but can run cool and therefore quiet. An A64-Sempron, A64 2800+, or maybe even a mobile Athlon XP would all be good choices.