Dumbledore could be a character in the fanfiction. Maybe he knew about the island all along and left it to Harry in his will. Or he could be a guiding spirit for Harry as he figures out what to do with the island.
There could be new characters too. For example, a hermit who has been living on the island secretly for years and has a lot of knowledge about the island's magic. Harry could meet this hermit and learn from him.
There could be some of the house - elves that Harry has freed. They might choose to live on the island, using their magic to help with various tasks such as farming magical plants or taking care of the island's infrastructure. Also, some of the students from Hogwarts who were inspired by Harry could be there. They might be learning new forms of magic that are specific to the island or studying the unique magical ecosystem there.
The fanfiction could be about Harry turning the island into a sanctuary for magical creatures. He might find some endangered or mistreated magical creatures and bring them to the island to live in peace. He'd then have to deal with the challenges of taking care of them all, like finding food sources and creating habitats suitable for each species.
His friends might be in it too. They could be the ones who join Harry on his yacht trips, having fun together. Maybe there's a character who is a bit of a troublemaker among his friends, always getting into funny situations on the yacht.
He might have discovered the island during one of his adventures and claimed it as his own. Since there was no one else on the island at the time, and he was the first to set foot on it and recognize its potential, he decided to make it his property.
Perhaps he uses the island as a secret training ground. He could practice advanced spells away from the prying eyes of Hogwarts faculty and students. Harry might also build his own magical fortress on the island for protection and privacy.
Perhaps it's an island with a hidden wizarding school, similar to Hogwarts but with its own distinct features. There might be ancient spells protecting the island, and the architecture could be a blend of different magical styles. Harry could have discovered this island during one of his adventures and decided to make it his own. He might use it as a sanctuary for those in need or as a place to study and preserve rare magic.
One possible plotline could be that Harry, through some magical accident or inheritance, comes to own Draco's family manor. Draco is then forced to interact with Harry on Harry's terms to regain his family's pride. Another could be that Harry wins a bet against Draco and the terms of the bet are such that Harry now has ownership over Draco's time, making Draco do tasks for Harry.
There could be a female leader on the island. She might be strong - willed and initially resistant to Harry's presence but later comes to respect him.
In this fanfic, perhaps Harry could turn the island into a magical sanctuary for endangered magical creatures. He might build a school there for young witches and wizards who are overlooked by the mainstream magical education system.
There's also a chance that Harry is a re - imagined version of an existing character within the same universe as LexCorp. The author might have given him a new backstory and set of circumstances that led to him owning LexCorp. For instance, he could be a former employee who had a grudge and through a series of events, ended up as the owner.