The Number Three 

In additional to the Torah verses about orlah, there are additional reasons why three years is a fitting age for a first haircut. 

The upsherin is the third in a series of "cuts" symbolizing a child's movement away from his mother and into the world. First the umbilical cord is cut after the birth, then the foreskin is cut during the bris. The haircut marks the beginning of the child's movement into society. At three, a child is ready to be less dependent on his mother and to interact more independently with other adults and friends.