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So all the demigods we know and love have all the obvious powers of their godly parents and some not so obvious powers. I'm pretty sure Poseidon only has power over saltwater, but that doesn't seem to stop Percy from controlling potions somehow. And basically every other domain that Poseidon has. I read a fanfiction once that said that since Percy was the first demigod kid in a while, he just absorbed all of Poseidon's aspects, which would make a lot of sense in retrospect.
But what about if he didn't?
What if the demigods inherited different aspects from their parents?
What if some daughter of Demeter had ice powers? Demeter did make winter, after all. And then they'd be able to make plants wither and not just grow. Or even make people wither, like how Nico turned someone into a skeleton once, although maybe not to that extent.
Apollo is the god of plagues as well as healing, so what about an Apollo kid who can make people sick? Imagine inflicting that on the titan army. You're fighting and then suddenly your enemies start sneezing like crazy.
What if some Hermes kids were just really good runners? Like, blink of an eye and they can sprint fifty meters or something. Hermes is supposed to be fast, after all, and he's also a patron of sports and sportsmen. He's sometimes called Hermes Enagonios, which means presiding over the games, and Hermes Promachos, which means champion.
Hermes is also a psychopomp who delivers dead souls to the underworld, so what if some of his kids have some death-related abilities? Like there'd be a scene where they think they're Hades kids for a second and freak out until Hermes claims them and now everyone's confused.
There was this one kid of Poseidon, Bellerophon, whose powers literally just involved being an equestrian.
Maybe to limit Percy's powers a bit and make him less overpowered compared to everyone else, he inherited more from the earthshaker and father of horses thing. The only time I remember Percy using his earthquake powers was the whole Mount St Helens incident. Would be interesting and ironic if Percy relied more on land-based powers considering he's a son of Poseidon.
What if Annabeth inherited less of the wisdom aspect? Athena is the patron of craftsmen and art too. Not to say that Annabeth isn't smart, but she's frustrated that her siblings are seen as better than her without even trying. So she just studies all the time to make sure that she keeps up with them and tempers her hubris more because she knows that she's much more likely to make a mistake. She's also really good at weaving and art, though Annabeth doesn't really do much art.
Aphrodite is a minor war goddess when she's Aphrodite Areia, which means war-like. Maybe Piper inherited more from that aspect of her while her siblings inherited more from the whole beauty and love part.
It's funny though, because Aphrodite is also known as Aphrodite Euploia, of fair voyage. Basically some people would ask her for safe travelling when they got on a boat. It'd be kind of funny if Piper just made the Argo II have a safer trip or something because it's a ship and she didn't even notice.
Hazel still keeps her gem powers because that's a curse, although we don't really see her using underworld powers. On a related note, Pluto is the god of riches, so it'd be kind of funny if Hazel's underworld powers included her being freakishly good at accounting.