Dumbledore could play a role too. He might be the one who gives Harry the idea in a subtle way. Maybe he sees the potential for Harry's cousin to be an important part of the wizarding community in the future, and by having Harry adopt him, he can ensure that the cousin is guided in the right direction.
One possible plot could be that Harry, having experienced so much loss and understanding the importance of family, decides to adopt his cousin. Maybe his cousin has also faced some difficult situations, like being neglected at home. Harry takes him in and starts to teach him about the wizarding world, creating a bond that was previously unthinkable between them.
Maybe Harry's magic gives his cousin a sense of security. If the cousin was always in some kind of danger, Harry could use magic to keep him safe. This would make the cousin less fearful and more adventurous. He could start to explore the world around him more, knowing that Harry has his back with magic. In addition, the cousin might develop an interest in magic himself, which could change his whole perspective on life.
In the fanfiction, Harry might raise his cousin in an attempt to break the cycle of animosity that has always been between them. He could show his cousin a different side of life, full of magic and wonder. Harry could also protect his cousin from the dark forces that are always lurking in the wizarding world, and in the process, his cousin could become a valuable ally to Harry.
Their parents' old friends could be main characters. They might come to check on Harry and his sister, bringing gifts and stories about their parents. These friends could offer guidance and protection when needed.
Well, in a 'harry raises his cousin fanfic', it could involve Harry having to take care of his cousin in a situation where the normal family structure has broken down. Maybe his aunt and uncle are unable to look after the cousin for some reason, like they've been put in a magical coma or something. Harry then has to step up and teach his cousin about the wizarding world, how to handle magic, and also about normal life things like school and making friends. He might face difficulties because his cousin could be resistant to the new world he's being introduced to, or Harry might struggle with his own insecurities about being a good caregiver.
It could be a dark wizard who has somehow managed to gain the power of Hollows. This antagonist would be extremely dangerous as they would combine the dark magic of the wizarding world with the destructive power of the Hollows. They might be after the cousin because of some special power or connection that the cousin has, which could be used to either rule both worlds or destroy them.
Well, clearly Harry is one of the main characters as he is the one taking care of the baby cousin. The baby cousin is also a main character, even if it's a young and not fully developed character yet.
Hermione could be there. She would be really helpful in planning and organizing the island. For example, she might set up a system for categorizing all the magical knowledge and resources on the island.
Ron could also be there. He might be initially scared of the dragon but then grows to like it. His loyalty to Harry would make him support Harry's decisions regarding the dragon, even if they seem risky. For example, when they need to hide the dragon from the Ministry of Magic, Ron would be right there with Harry, coming up with crazy but effective plans.
There could be a local wizard who becomes Harry's mentor in a way. Since Harry is on his own, he might seek guidance from someone outside of his usual circle, like an old hermit - like wizard who has a lot of knowledge about the more hidden parts of the wizarding world.