Question: I want it to perform good enough as a multimedia machine. Should I get one that is fan cooled or passively cooled? I think I want one with an HDMI port.

What chipset should I look for?

Answer: 
I agree with an Ion chipset, there's really no competition in the quiet/ITX space. I've got one of [these](http://www.newegg.com/Product/Product.aspx?Item=N82E16813500035&cm_re=zotac-_-13-500-035-_-Product) powering a secondary HTPC for my bedroom. It has a [Celeron 430](http://www.newegg.com/Product/Product.aspx?Item=N82E16819116039) 35W chip in it, and it runs good and quiet. HD playback is no problem, once I got a codec that uses the hardware decoder instead of trying to shove it through the CPU.

My original thought was to go with the Atom 330 version of this board, but what I've seen about it lead me to believe that it would be too sluggish when using the VMC interface. 4 l-cores or not, its still an in-order architecture. If silence is more important to you, there are some Atom Zotac boards that are passively cooled, and have a power brick instead of a normal PSU.
+1 for an ion as a multimedia machine. low power (both CPU and Wattage) but with a graphics chipset that can hadnle FULL 1080p HD.

some even come with an HDMI port.

If you want a nice easy answer, some pre built ion nettops exist such as the Acer revo, or asus ionstar