The hardware side of this project is easy. The hard part is putting together a small Linux OS for it and customizing LinuxBIOS for this kind of system. If any one wants to start such project let me know. All I have are ideas, what I don't have is the knowledge to pull this off.
What hardware do you have in mind? To me the software of this actually seems like the easy part. There are loads of ways to make a small customized linux image.
http://www.emdebian.org/ http://familiar.handhelds.org/ http://www.linuxfromscratch.org/ http://openwrt.org/ http://www.damnsmalllinux.org/ http://www.pengutronix.de/software/ptxdist_en.html
Just to get started....
Not quite so small but has loads of stuff you would need..
http://www.geexbox.org/en/index.html
Horsepower wise though you are really going to have some beef to compete with the graphics power of the Xbox or ps2 line of products. That means the later nvidia or ATI cards.
A gaming console is also going to need some sort of low noise, smaller type case setup.
Neither of the above is cheap. $300 is a would be a pretty good deal for what you get if it wern't for the stupid DRM they use. Although I think ps2 has linux dev kit for it dosen't it?