One idea could be that Percy discovers a demigod child who has been abandoned in the mortal world. He decides to adopt the child to protect them from monsters and to give them a proper home at Camp Half - Blood. He then has to teach the child about their demigod heritage and how to use their powers.
Perhaps the child is the offspring of a Titan. This would make for an interesting story as Percy would have to deal with the potential danger and mistrust from other demigods. But Percy, being Percy, would see the good in the child and adopt them regardless. The child might have unique powers that set them apart from other demigods, like the ability to control time or space.
Maybe the child is an orphan with no idea they are a demigod. Percy senses something special about them and decides to adopt them. He then has to deal with the challenges of raising a child while also protecting them from the various threats in the Percy Jackson universe. For example, he'd have to train the child to fight and also keep them hidden from the wrong eyes. He could introduce the child to Camp Half - Blood, where they meet other demigods and start to discover their true identity.
One challenge could be dealing with the child's past traumas. If the child has been through a lot, like being orphaned or facing monster attacks on their own, Percy would have to help them heal. He's not a trained psychologist, so it would be difficult for him to know exactly how to handle it.
One possible plot could be Percy finding Thalia in a difficult situation, like being hunted by some monsters. He decides to take her in and protect her. He teaches her about the demigod world and they form a strong bond as siblings. They train together to become stronger and face various challenges from the gods and other mythical creatures.
She could be the daughter of a minor god that has been forgotten or shunned. For example, the daughter of Hecate's lesser - known cousin who has some magic abilities but no real place in the demigod world until Percy takes her in.
One possibility is that Percy discovers a young demigod girl in a dangerous situation, like being chased by monsters. He decides to take her under his wing and protect her. They then embark on a journey to Camp Half - Blood, facing various challenges along the way.
One great 'child percy jackson fanfiction' is 'Percy Jackson and the Quest for the Lost Artifact'. It has an exciting plot where Percy and his friends embark on a new adventure outside of the main series' storylines. The characters are well - written and stay true to their original personalities.
In this kind of fanfiction, Percy could introduce Harry to a whole new world of Greek mythology. Harry might learn about demigods and their powers. Percy could help Harry better understand his own magic in a different way, maybe by comparing it to the powers of the demigods. Also, Harry would probably be introduced to Camp Half - Blood and make new friends there.
In this fanfiction, Voldemort or his remaining followers could be major antagonists. They wouldn't like the idea of Harry having new allies in Percy and the demigod community. From the Percy Jackson side, the Titans or some of the more malevolent gods might not be happy with the alliance either. They could try to break it up or cause trouble for the two boys, seeing them as a potential force against their own plans.