What helped me when I was a kid were games like Quake 3 Arena, Worms, and Mechwarrior 2, 3, and 4.

Q3A was because of the speed. you had to be fast with the mouse if you hoped to compete. Add to the fact that the bots in Q3A, at the time anyways, were quite good. you can play it offline with bots or even over a local LAN.

Again going back to a LAN staple but Worms is also good. you need precision with the mouse on that one. lots of geometry at play.

Finally the Mechwarrior games which really emphasize the mouse and keyboard combination. Torso Twisting and Flicking, positioning of your legs in regards to your torso, etc. really helps with mouse coordination.