The Mysterious Old Man is a man who can teleport you to the Maze and Mimerandom event. In the Maze random event you must complete the labyrinth as quickly as possible to ensure a better reward. When underwater the Mysterious Old Man will appear with diving apparatus and a fishbowl helmet.
He used to mark the start of many random events, appearing and giving the player a puzzle box or teleporting them off to the Maze or just thanking them for talking to him. The puzzle box is a spinning cube with a number on each side, inside of a shape. The rewards from this random can be very generous, as you can get a half key from the Mysterious prize box.
Although the Mysterious Old Man can still give optional random events, he can also be found upstairs in a house in eastern Falador, saying he is trying to solve a puzzle-cube, and also that he is retired.