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.
Percy would also face challenges within Camp Half - Blood. Some of the other demigods might not be happy about a new child in the camp, especially if resources are scarce. He would have to deal with any jealousy or resentment among the campers while also making sure the child feels welcome and part of the community.
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 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.
The child would likely face a lot of prejudice. In the Percy Jackson universe, the idea of Artemis having a child goes against her long - held status as a virgin goddess. So, many characters might look down on the child or be suspicious of it. Also, the child would have to deal with very different powers from its parents. Percy has powers related to the sea, and Artemis has powers related to the moon and the hunt. Combining and controlling these different powers could be a huge challenge for the child.
He'd probably have trouble coordinating the two halves and dealing with their different thoughts and desires.
In a 'Percy Jackson adopts Harry Potter' fanfic, the plot could start with Percy being on some kind of mission that crosses into the wizarding world. He stumbles upon Harry, who is alone and vulnerable. Percy decides to adopt Harry because he sees the same kind of determination and courage in Harry that he has. As they journey together, they exchange knowledge. Percy teaches Harry about Greek mythology and combat skills, while Harry shares his knowledge of magic. They form a bond that helps them face threats from both magical and mythological creatures, and they might even team up to fight against a common enemy that is a threat to both their worlds.
I think a character with the ability to control time would be really interesting. Imagine Percy having to deal with a demigod who can accidentally mess up the time continuum. It would create all sorts of wacky situations at Camp Half - Blood. Like, one moment they're having breakfast, and the next, it's suddenly midnight because the adopted demigod made a time - related mistake.
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.
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.
Perhaps he has been exposed to some sort of deadly and unknown substance in the mythical world. This could be a new type of monster venom or a strange plant that he accidentally ingested. The substance could be attacking his body from the inside, causing his internal organs to bleed, and thus he throws up blood.