Age of Empires 2 with custom settings. I believe you can set star cash way up and instant build. We used to play those games at lunch during high school and had a blast doing it (even the physics teacher joined us).
C&C Generals, if you can all run it, is an amazing co-op experience. Once WMD's hit the field the game gets very interesting and airstrikes make the game crazy throughout.
Quake in any form is great for co-op, I was always partial to Quake 3: Rocket Arena.
For lower specs I would also suggest some Team Fortress Classic, Counter Strike 1.6 (Source is preferrable and not that graphically demanding), Unreal Tournament at whatever range of graphics you can play at the original is still amazing.
HL mods offer a large range of game types from Natural Selection to Sven Co-op. Just get steam and buy a bunch of cheap old games...
Hell, I still pop in Warcraft 2 on occasion. The custom map tools are easy and you can make some fun stuff even though the gameplay is dated. Starcraft would be a better option if for no other reason than it's more popular.