Miner A.J. was a kid who went prospecting for gold in the Grand Canyon during the days of the gold rush.
He looks like A.J. dressed as a gold prospector.
In Breakin' Da Rules, Timmy travels to the time of the gold rush to look for Da Rules and finds it at the bottom of the Grand Canyon, where Miner A.J. is also trapped. To clear the level, Timmy has to climb down the canyon and rescue him.
At the end of the game, if the player collected all the magic dust, they are treated to a bonus cutscene showing that, sometime after the rescue, Miner A.J. got trapped in the canyon again and tries to climb his way out. During his attempt to climb out, some Magic Dust gets swept over the cliff by Jorgen Von Strangle and knocks him down again.