In Lord of the Flies by Golding, what do Simon, Piggy, and the boy with the birthmark all have in common?

There isn't really a personality trait, physical appearance, or function in the plot that these three boys share. What they do share is the fact that all three of them die, and they are the only ones whose deaths we can be certain of. Through their deaths, they may also represent the "loss of innocence" theme that runs throughout the story.

The boy with the birthmark is a "littlun," and is the first to speak of the Beast. We don't see his death or have it confirmed, but he is never seen again after the fire in Chapter 2, and it can be assumed that he died in it. In his case, his innocence lies in his age and his inability to think rationally; he is responsible for conjuring the idea of the Beast, and the fact that he is the first to die almost seems like the fulfillment of a curse for bringing this supernatural fear to the group. There may also be a more far-reaching connection to the power and danger of prophecy, as all three boys are, in their own ways, responsible for bringing knowledge and portents to the group.

Simon dies mostly due to bad luck; he comes out of the forest at the climax of a thunderstorm and a ritualistic "pig hunt" initiated by the near-savage hunters of Jack's group, and they supposedly mistake him for the Beast. Simon is also the most obviously prophetic character, due to his conversation with the Lord of the Flies, his statement that Ralph will get off the island someday, and his "fits."

Piggy dies horribly and intentionally, at Roger's hand amid the jeering of the Castle Rock tribe. His death, being the most intentional, may have been "saved" for last by Golding in order to demonstrate the boys' diminishing restraint. It is telling that the conch perishes with Piggy, simultaneously destroying both his superior mind and the semblance of order on the island.