I hate the emphasis that Percy Jackson puts on bloodlines. If you have a godly parent, you get to be a hero. If you don't, tough luck!
And what I find so weird is that Greek mythology is not like this. Sure, you have Heracles, Perseus, and Theseus being sons of big name gods; but you also have Odysseus, Patroclus, Diomedes, Atalanta, Menelaus, Hector, Jason, both Ajaxes, Cadmus, Pelops, and many others who have no gods as parents.
Plus, there's a grey area. Achilles is the Greek hero per excellence, incredibly skilled and agile—and his mother was a sea nymph. He wasn't the son of Zeus, Poseidon, Apollo or any other of the "important" gods, and he was still the greatest of his generation. Orpheus is attested as the son of Apollo in a few sources, but he was mostly regarded as the son of a mortal and the muse Calliope; again, not a big shot god, a minor goddess.
If Greek mythology followed the rules of Percy Jackson, only Heracles, Perseus, maybe Theseus and Bellerophon, depending on which of their parentages you believe, and a few others would be heroes; everyone else would be a footnote. And I ask, why include this weird bioessentialism in a children's series? Why couldn't Annabeth, Grover, Nico, or any of the main supporting cast members just be an extraordinary mortal, like Odysseus and so many others? This is like if in Narnia only children born of magical parents could pass through the closet