(I tend toward puzzle games and those friendly for my kids versus the latest high graphic games)
These games all worked well on my old Dell laptop with 1.3 GHz processor and 748 Mb Ram
SuperTux & SuperTux2 - the game my 7 year old has played more than any other
Pengus - a Lemmings like game of saving the penguins
FreeCiv - based on the code from Civilization 2, a turn based strategy game. comes with 3 or more different front ends to suite your graphics tastes.
Frozen Bubble- A great little puzzle/arcade game my daughter loves to play and make custom levels.
TuxMath & TuxTyping - top of the line fun educational games in multiple languages
Secret Maryo Chronicles ----
Gnome Mastermind - Mimics the classic board game of the same name but you can play with one player. A favorite logic game of mine. Don't need Gnome installed (I'm using on LXDE Lubuntu).