Do demigod powers get handed down? In Harry Potter, wizarding blood and magical ability is handed down, sometimes even if a wizard marries a Muggle (I was about to say mortal :) )
Magic can skip a generation.
So what if a demigod marries a mortal? Would the child have demigod symptoms, such as ADHD/dyslexia etc?
Magic in Percy Jackson universe isn't even necessarily genetic in the first place.
This is shown by the fact that children of Athena aren't born genetically from her, and yet still inherit her gifts.
(I will leave aside the actual reproductive biology of the original Greek mythology, where the childbirth can happen from dolphins, bulls, or golden rain).
Having said that, Legacies (demigod descendants) in the sequel trilogy do have some level of power, though it is diminished compared to demigods.
Q: If two demigods had a child, would that child be a quarter-blood, a demigod, or what?
A: ... If they did have children, the kids would probably pass for normal mortals, since the godly powers get diluted with each generation. If the parents were extremely strong, the child might be more like a demigod. At Camp Jupiter, things are a bit different, as you know if you've read The Heroes of Olympus.
(Riordan's FAQ)
However, some unique powers can be inherited; for example Frank Zhang's power of turning into animals that got transmitted from parent to child all the way from ancient Greek demigod who was a son of Poseidon.
I don't think this is addressed in canon, but consider the case of Frank Zhang (first appears in Son of Neptune). His father is Mars (Roman form of Ares), and his mother is a distant descendant of Poseidon. His ancestor Periclymenus, one of the Argonauts, who is himself a grandson of Poseidon. Poseidon gave him a special gift of being able to transform into any animal, and that it would be hereditary.
Because there is no mention of any other powers from Poseidon, we can conclude that demigod powers aren't usually hereditary.
