Perhaps the story could focus on Percy's child trying to live a normal life in the mortal world but constantly being drawn back to the world of the demigods. They could struggle with fitting in at school while also having to deal with strange occurrences that only they seem to notice, like seeing mythical creatures in the corner of their eye. This internal conflict between two worlds could drive the plot forward.
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 Percy leading a new generation of demigods in a battle against an ancient evil that has been awakened in the future. Maybe he has to train them in new ways to combat the threats.
One plot idea could be that Percy's singing has magical properties. Maybe it can heal the wounded or calm the angry sea gods. So, he has to use his singing to save the day during a big crisis in the demigod world.
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.
A great plot could be that the children are sent on a quest to a new, mysterious location that was previously unknown in the Percy Jackson universe. They might have to deal with strange new monsters and make allies among the local mythical beings. For example, they could end up in a hidden valley where the spirits of nature are at war, and they have to figure out how to restore balance. This would involve them using their powers in new and creative ways, just like their parents did in the original series.
One idea could be that the child has unique powers inherited from both Percy and Artemis. For example, the ability to control water like Percy and also have a strong connection with the moon and hunting like Artemis. Another plot could involve the child trying to find their place in the demigod world, being part of two very different lineages. They might face challenges from both sides, some who are suspicious of their mixed heritage.
A really interesting plot could be Percy losing his memories due to a curse. His friends would have to help him regain his identity while also protecting him from enemies who see this as an opportunity to take him out. As he slowly remembers bits and pieces, he uncovers a conspiracy within the Olympian ranks.
One plot idea could be that Percy, while on a quest, stumbles upon Hestia's hidden sanctuary. There, he begins to understand her in a new way and feelings start to develop. Another could be that Hestia, seeing Percy's selfless acts, becomes intrigued and starts to interact with him more, leading to a romantic connection.
One idea could be that Percy's child discovers they have special powers like Percy. For example, they can control water in unique ways. Another could be the child getting into trouble at a new school for demigods and Percy having to step in to help.
In a Percy Jackson has a Omnitrix fanfiction, the plot could involve Percy being sent on a special mission by Chiron. During this mission, he stumbles upon the Omnitrix. He soon discovers that the Omnitrix has some sort of connection to the Greek gods. As he uses it to transform into various alien forms, he finds that each form has a unique ability that can either help or hinder him in his quest. For example, one form might be able to swim super fast in the ocean, which is great for underwater battles, but another form might be weak against fire, which could be a problem when facing a fire - breathing monster. Along the way, he has to learn to balance his use of the Omnitrix with his own demigod skills and knowledge of Greek mythology to succeed in his mission.