By now we should all know that the giant statue on the island is Tawaret, the Egyptian goddess of fertility and childbirth. This is interesting because it is well known that women are unable to give birth on the island. This was not the case, however, in the 1970's when Amy gave birth to Ethan. So between then and now, something must have happened to make it so that women could not give birth.
When Sun asked Ben what happened to the rest of the statue, Ben said it was like that when he got there, but Sun could tell he was lying. This means that when Ben was on the island, back in the 70's, the statue was still standing. I believe that the statue had some sort of power that allowed birth on the island. Once it fell, sometime between '77 and'04, the power was lost and the island became almost completely unfertile. Could it have been the incident/hydrogen bomb that caused it to collapse? Perhaps Claire was able to give birth because the foot of the statue remained and it provided a small amount of its powers.
Also, the statue was holding anks, symbols of eternal life, so that may provide some clues about Richard and Jacob.