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 this fanfiction, Harry's magic might influence his cousin in a very personal way. It could be that Harry uses magic to heal some emotional wounds of his cousin. For instance, if the cousin has some past traumas related to family or school, Harry's magic could be used in a way to soothe those pains. This would lead to the cousin becoming a more well - rounded and positive person. He may start to see the good in others and in life in general because of the positive impact of Harry's magic.
His magic could make the Queen see him as a valuable asset. She might rely on his magical abilities for protection or for solving problems that normal means can't handle.
Some of his friends might be very supportive. Hermione, for example, would probably offer a lot of practical advice on how to raise a child properly. Ron, on the other hand, might be a bit more hesitant at first but would come around. Overall, they would likely become closer as they help Harry with this new responsibility.
It could change the story in many ways. For one, Harry might have more of a reason to stay closer to home. Instead of always being eager to rush back to Hogwarts, he might feel a sense of responsibility towards his baby cousin. This could lead to new storylines where he has to balance his wizarding life and his family 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.
The magic within Harry could be influenced by the raw power of the giants. He may be able to access a deeper well of magic due to his exposure to their strength. His spells could be more powerful but also less refined in the beginning. As he enters the wizarding world, he has to learn to control this power better and mold it into the more accepted forms of magic used in Hogwarts.
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.
Yes, there are likely fanfictions out there with such a concept. Fanfiction writers often like to create crossovers between different popular universes like Percy Jackson and Harry Potter. It's a fun way to combine the magic of Hogwarts with the demigod world of Percy Jackson.
In some fanfictions, it could be through intense emotional experiences. For example, when he faces a great loss or discovers a deep truth about his family, it triggers something within him and unblocks his magic.
Since Madame Maxime would expose Harry to different magical cultures, his magic could develop in a more diverse way. He could have a better understanding of magic related to magical creatures, like using magic to communicate with veelas or control giants. This would be different from the more traditional Hogwarts curriculum that focuses mainly on spells for general use. His magic could also be more attuned to the environment of the French wizarding world, which might have different ley lines or sources of magical energy.
Harry's magic could be stronger. Dorea's influence and knowledge of magic could have allowed Harry to tap into his magic potential more easily. Maybe he would be able to perform advanced spells earlier. Also, his magic might have a different 'flavor' so to speak. Since Dorea comes from a different magical background than the Dursleys, Harry's magic could be more attuned to the traditional wizarding ways of magic use.