What is the purpose of an artist's signature on an old canvas painting?

Prospective fame and notoriety is the purpose of artists signing their work. Signing paintings started as a way of expanding the potential pool of sponsors.

International trade grew exponentially during the Renaissance. Artists knew that their works had the potential to travel great distances, so they began signing their paintings quite literally so that people would know, and hopefully remember, who had created them. Many artists relied heavily on wealthy sponsors who, at the time, hired famous individuals to paint for them. The more one's name was circulated in the art world, the more well-known he became. In contemporary times, signing one's name is more a method of authenticating a work.