Try searching for Roguelike games. The obvious one is Rogue, but ADoM (Ancient Domains of Mystery) is a similar improvement, and there ar emany others in teh genre. These are designed to run on all sorts of systems so shouldn't have any problems with running them.
I know a great game called MechQuest. It's made by artix entertainment and is a great turn-based game. also, no system requirements, it's a browser run game.
My favorite Dos games were Quake, Doom, and Jill of the Jungle.
In ye olde days many popular Dos games switched to or were connected to "Shareware". Usually there was a Demo version that you could Shareware so friends could try the game. That might help you find them on things like eBay or online.
Remember your Hard Drive size is going to be something you'll want to keep an eye on.
