Perhaps Percy's immortality makes him an outcast at Westover Hall. The other students are either jealous or scared of him. He then has to find a way to fit in while also hiding his true nature from the teachers who might be working for the enemies of the gods. He could start a secret club with a few students who he trusts to help him in his quest to understand more about his immortality.
One idea could be that Percy discovers an ancient immortality spell hidden in the depths of Westover Hall. He has to face off against some dark forces that also want the spell for their own evil purposes. Maybe he teams up with some new friends he makes at the hall to keep the spell out of the wrong hands.
Well, in this fanfiction, Percy's immortality could lead to some really interesting character development. He may start to feel lonely as he realizes that he will outlive all of his friends. This could cause him to be more withdrawn at first. But then, as he interacts with the students at Westover Hall, he might find new meaning in his immortality. He could start to see it as a way to protect others not just for a short time, but for as long as he exists. He could also form deep and lasting friendships with those who accept him for who he is, regardless of his immortality.
There might be a fanfiction where an older Percy Jackson goes to Westover Hall undercover to investigate some strange magical occurrences. He teams up with some of the students who have a latent magical ability. In this story, he has to hide his identity while trying to figure out what's going on. The author could play with the idea of Percy using his skills in a more mature and strategic way compared to when he was younger.
One interesting plot could be Percy and his friends having to solve a mystery within Westover Hall that involves an ancient artifact hidden there. Maybe it's a lost weapon of a Greek god that has the power to control the weather. Another plot could be a new character arriving at Westover Hall who is secretly a demigod in disguise, and Percy has to figure out their true identity before a monster attacks. And there could also be a plot where the school is hosting a competition, but it's secretly a test by the gods to see which demigod is the bravest and most worthy.
You can start by looking on fanfiction websites like Archive of Our Own (AO3). They have a large collection of Percy Jackson fanfictions, and you can use the search function to specifically look for ones related to older Percy and Westover Hall. Just type in 'older Percy Jackson Westover Hall' in the search bar.
Percy himself is always an intriguing character to include. His sarcastic humor and his ability to get into trouble and out of it are fun to write about. Annabeth is also great. Her intelligence and her strategic mind can add a lot to the story. Grover is another one. His loyalty and his goat - like tendencies can create some humorous and heartwarming moments. And don't forget about the villains. A well - written villain like a powerful monster or a devious demigod who has a vendetta against Percy can make the story really exciting.
One idea could be Percy exploring different mythologies in his immortal state. Since he has a connection to Greek mythology, he could now travel to other realms related to Norse or Egyptian mythology and interact with their deities. Another could be Percy dealing with the concept of eternal loneliness. He has seen his friends and family age and pass away while he remains immortal.
One idea could be that Percy discovers a hidden power within himself that makes him an even more powerful immortal warrior. Maybe he can control time briefly, allowing him to outmaneuver his enemies in battles. Another could be that he teams up with an ancient immortal warrior from a different mythology, like a Valkyrie from Norse mythology, and they have to combine their powers to defeat a new threat to the world of demigods. And perhaps there's a prophecy that only an immortal Percy can fulfill, which involves traveling to the Underworld and making a deal with Hades to gain an ultimate weapon.
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.