Cheap?
You have to take a few things into account.
PS2s do support EDTV.
EDTV is 720p but essentially 1080i HDLite watered down.
I recommend getting the ps2 component cable, its basically 5 wires...3 for video, 2 for sound.
If youre just going to use this for the PS2 then it doesnt matter what inputs the LCD has, you can pick up a Component to DVI or RGB converter for about $20.
Its even better if the LCD has Component input natively.
The best units are made by Samsung and Viewsonic(surprisingly).
I dont know the failure rate of Viewsonics. There is a nice Viewsonic that you can pick up pretty cheap. Try checking newegg.com. I recommend Samsung since I have never had a problem with them. Since your using this as a gaming screen, make sure you get an LCD with a low response time.
CRTs have immeasurable response times since their so fast and great for gaming and imaging, but since you want an LCD.......the lowest response time Ive seen is 8ms.
8ms should be ok for games. If you get a 16 or 32ms screen, you'll see that games start to look like gelatin.
Also, many people get problems using component with PS2
..you have to set the PS2 to component output in the menu, switch plugs and then reboot the PS2. I have noticed that the PS2 doesnt broadcast Component by detection/plug+play.
Component <-> DVI
http://sewelldirect.com/dvivideoadapter.asp
Component <-> RGB
http://www2.dvigear.com/vgahtocovish.html
(with this option you will need female-female sleeve to connect 2 male connetors unless you find a convert like first one)
PS2 Component Cable
http://www.redoctane.com/componentcableps2.html
(you can probably get a better cable somewhere else)
Composite, S-video, and coaxial deliver poor signal quality and using these on an lcd is pointless.
(you can probably get a better cable somewhere else)