One idea could be that the new girl is a demigod from a very rare and powerful lineage. For example, she could be a descendant of a primordial god that no one has seen in centuries. This would bring a whole new dynamic to Camp Half - Blood as everyone tries to figure out her powers and where she fits in.
Maybe the new girl has a special connection to Percy. Like she has visions that are related to his future or past. This could lead to an interesting adventure where they have to figure out what these visions mean and how to use them to prevent some impending doom. It could also create some tension with Annabeth as she might be a bit jealous of the new girl's connection to Percy.
The new girl could be an undercover agent sent by an enemy of the camp. She pretends to be a lost demigod but in reality, she is there to gather information. However, as she spends time at Camp Half - Blood, she starts to question her mission and might end up switching sides. This would involve a lot of mystery and intrigue in the fanfiction.
She could be the daughter of a minor god that hasn't been introduced much in the original Percy Jackson series. For instance, the daughter of Hecate. This would give her powers related to magic, which would be really cool in the context of the story. She could use her magic to help Percy and his friends in battles or to solve some of the mysteries they encounter.
One plot idea could be that Percy Jackson somehow ends up in the world of 'New Girl' and has to navigate the apartment - sharing and relationship drama while also hiding his demigod powers. For example, he might use his water - controlling abilities in a creative way to solve a plumbing problem in the loft, all while trying not to arouse suspicion.
Imagine a fanfic where Percy has to navigate the New York City traffic while being chased by a pack of hellhounds. He has to use his water - based powers in creative ways, like creating water slides on the streets or using sewer water to slow down the hellhounds. This would be a really action - packed and fun story.
A plot could be that Percy is sent on a quest that leads him to the 'New Girl' universe. He has to find a magical item hidden in the loft. The 'New Girl' characters think he's crazy at first, but as they see the magic unfold, they help him. Another plot could be that one of the 'New Girl' characters discovers they are a demigod. Jess, for instance, realizes she has demigod powers and Percy has to train her. This would bring a whole new dynamic to the loft as she tries to balance her normal life and her new demigod life.
She could be really shy at first. Maybe she comes from a place where she was always bullied for being different, so when she arrives at Camp Half - Blood, she has a hard time opening up to others. But as she discovers her powers and makes friends, she slowly becomes more confident.
Perhaps a fanfiction where the New Gods create a sort of competition for demigods. Percy enters this competition and discovers that there are rules he's not used to, and he has to use his wits and skills in new ways to win. And along the way, he makes some unexpected allies among the new gods.
Another possibility is that Percy is sent to Hogwarts undercover as a Hufflepuff student to investigate some strange magical activity that might be related to the Titans or other Greek monsters. He has to keep his identity as a demigod a secret while trying to fit in with the Hufflepuff house. He could learn a lot from the Hufflepuff way of life, like their patience and kindness, and in turn, he could teach them some things about courage in the face of danger that's different from what they're used to in the wizarding world.
In an Alaska - based Percy Jackson fanfiction, Percy could be on a quest that leads him to the frozen tundra. He might discover an ancient Greek artifact that has been hidden there for centuries. This artifact could hold the key to preventing a major disaster, like the awakening of an old and powerful enemy. Along the way, he could team up with some local Inuit demigods who have their own unique powers and knowledge of the Alaskan wilderness.
One idea could be that Percy uses his demigod skills in the courtroom. For example, his ability to read Ancient Greek could come in handy when dealing with old legal documents or in cases related to archaeology where Greek inscriptions are involved. He might also use his combat skills in a more metaphorical way, like being a fierce advocate who 'fights' for his clients.
One interesting crossover could be Percy Jackson finding himself in the world of 'New Girl'. He might use his demigod skills to solve the zany roommate problems. For example, when Nick and Schmidt have one of their epic arguments, Percy could use his water powers to cool them down, literally. Maybe he'd also introduce the 'New Girl' characters to some of his demigod friends like Annabeth, and they'd all go on adventures in Los Angeles together.