>>142592I'd also look at strategy games, but that's mostly because they're my preference.
Battle for Wesnoth is the perennial "Bad computer wat do" answer and it's pretty nice.
Master of Orion II is still the absolute best space 4x game of all time, bar none, and you can play that on a fridge nowadays.
Civ 3 was a personal favorite of mine for a long time. Mechanics are a bit outdated by now but there were some really cool and very creative mods.
If you want to jump into the deep side of the pool, Dominions 4 is a pretty crazily complex game with an insane amount of options and a decently active but small community. Apparently it runs well even on old hardware, but you might have a problem viewing some of the battles.