I suggest you to play turn-based strategy games (TBS), most of them have hotseat multiplayer. Heroes of Might and Magic III, as I remember, was ported to Linux -- though you will be limited to only the first version, no addons (if I am not mistaken). Anyway, the Windows version works well under Wine even on Asus EeePC series laptops.
If you decide to play under Wine, definitely try Heroes of Might and Magic IV, for many reasons its balance is objectively more diverse than in HoMM III, more like that of HoMM II, and far less rules are needed to make gameplay more fair.
For example, HoMM IV tournaments usually only ban one thing, one hero class (the Thief), and it is actually allowed on the smaller maps sometimes. For HoMM III the tournament lists of prohibited options take pages and pages, because too many options allow you to break the game.
Another good TBS to try is Age of Wonders series. My favourite is the first one, made back in 1999, but you might prefer some other one, so try them all. The first game runs perfectly under Wine, Age of Wonders III has native Linux support, and I have no information about the second game in the series.