The Greek gods were sufficiently advanced aliens with access to powerful technology. In the begining of our worlds formation, Zeus created mankind from his own flesh. These mortals were made in their god's image, as sons and daughters of the gods. However, Zeus had a bad habit of "spreading his Royal oats" a little too often. This led to a trail of children across the planet.
The children of Zeus number in the hundreds. However, the vast majority of them are generic mortal humans with no special properties to them. Some children are chosen as babies by a goddess, who takes an interest in them. She nurses the child for six months, after which they complete their transformation into a full demigod.
Breast milk contains antibodies that help your baby fight off viruses and bacteria, as well as fend off certain diseases. How can this affect a child's development and lead to a demigods transformation?