The idea that rubbing a Buddha statue's belly brings good luck is a piece of folklore that apparently derived from the tradition of the Laughing Buddha. The Laughing Buddha is based on a Zen monk from over 1,000 years ago.
According to Religion Facts, the Laughing Buddha's image is meant to convey abundance and contentment. He is always shown laughing with his large belly exposed as a symbol of happiness, plenty and good luck. Often Laughing Buddha statues include begging bowls, children and a large sack, presumably filled with good things. He is the patron of restaurateurs, bartenders, the weak, the poor and children. Tradition holds that rubbing the Laughing Buddha's belly brings good luck and prosperity.