On that end, I assume you'll be thinking of some sort of HiDef connection. There are now graphics cards that have HDMI outputs, which would be the best route to go for ACPI compliance. You'll definitely want a passively cooled GPU here, so options will be limited.
As for capture hardware, I would recommend a PVR150 MCE, or one of Hauppauge's higher models if you would like to record more than on channel at once. MCE 2005 will allow you to record HDTV, but only terestial HD signals.
Since MCE records in a proprietary MPEG-2 format, all your recordings will take up a lot of space, usually about 2.66GB per hour of television recorded (in standard def). On that note, I highly recommend using a separate hard drive for your recording, preferably 200GB or larger. Use Tweak XP you move your "My Documents" folder over to the recording drive.
You'll also need an MPEG-2 software decoder, such as Nvidia's PureVideo or ATI's variation of the same. Cleve produced a nice article here on Tom's comparing the two; I suggest you read it.
As for ram, if this is a dedicated HTPC that will be nothing more than a glorified TiVo, 512MB of ram will be fine.
Motherboard will matter little, but a MicroATX might be preferred do to the possibility of using a small case. Some more important points are effective CPU fan control in the BIOS, and of course vcore options that allow you to undervolt. I had tremendous luck in that regard in the BIOS options on a socket 754 T-Force 6100, which would be a great choice if you don't mind buying old (but still 100% effective) tech. Using it I was able to undervolt a Sempron 2800 to 1.1 vcore, the point at which the cpu fan was no longer needed on the stock heatsink. The T-Force 6100 is also made in the socket 939 and (I believe) AM2 format, so they will probably have similar underclocking/fan control feature as the socket 754 board I used in the past. One thing I'm not sure about on those T-Force boards is their ability to suspend to ram (S3 standby). You'll definitely want a functioning S3 for your Media Center PC, as it cannot wake itself from hibernate to record. You might want to check with SilentPCreview's forums for a great mobo recommendation. While you're there, check out their PSU reviews to get an ultra quiet power supply.
MCE 2005, to the best of my knowledge, can't really utilize a dual core cpu, so a single core is the way to go. I recommend an Athlon 64 3000 socket 939, preferably undervolted so it generates as little heat as possible. It's an economical processor that is extremely effective for an MCE HTPC.
A build along these lines should be easy to achieve under $800. Hope that helps get you started. Good luck.