The boulder can only be moved by the Golden Child at a random point following its appearance. The Golden Child is a magical baby born to a couple late in the game. Its birth requires several in-game conditions to be fulfilled, so it is often one of the last puzzles the player solves.
Four conditions must be met before the Golden Child can appear. First, the lagoon must be filled; second, the idol must be completed; third, the village's fertility level must be at three; and fourth, one of the women in the village must be nursing a baby. To get the Golden Child, click on the nursing mother and drag her, along with the baby, into the lagoon. The mother will swim around for a bit with the baby and then come out of the water. When the baby has finished nursing, it will appear in the village as the Golden Child.
The Golden Child can perform several different special tasks, including boosting village production and speeding up work. However, it usually can be found running around the village like any other child born in the game. Shortly after birth, the Golden Child will approach the boulder and move it magically, allowing access to the cave behind. This event is random and there is nothing the player can do to make it happen more quickly.