In addition to all the recommendations here, I'd say Red Faction is also good. The multiplayer on the game is pretty fun, particularly with the deformable terrain, and it's cheap; $10 at Wallymart. I'm not sure if it has an option to play multiplayer without a disk, but again, it's cheap and fun.
Other possibilites if your kids are adventurous at all are Tribes and Tribes 2, though I recommend using mods rather than the base game. You can check out planettribes.com for details, even links to mods and stuff. I recommend Shifter since that's what I play, but I'm kinda biased. Tribes and Tribes 2 are both cheap, though Tribes 2 may require a large-sized patch to run and Tribes may be hard to find. Both are a bit of a trip to get used to since players have jetpacks in the game.
The orginial Soldier of Fortune is good as well. Kept my dorm buddies and I busy for hours and hours. Should be cheap as well. But it, like its sequel, is pretty bloody. If you're worried about that kind of thing make sure the parental controls are on. They'll complain a little but trust me, the game is fun enough they won't care.
Oh yes, I have Unreal II but not Unreal Tourney 2k3. I'm assuming it has all the same weapons as the singleplayer game? If so that will be high on the list of games to try. The Flamethrower alone makes it worth playing. Burn, burn! Hehe.
P.S. All those are First Person Shooters. For non-FPS games I'd say any of the Worms games are high on the list. Humorous and entertaining!