In some fanfiction, Harry might renounce his name to explore a new identity. He has been through so much as Harry Potter, and by giving up his name, he can break free from the predefined roles and destinies associated with it. He could then embark on a journey of self - discovery, free from the constraints of the name that has both defined and haunted him.
One possible reason could be a fallout with his friends in Gryffindor. Maybe he feels betrayed by them in some way in the fanfiction world. For example, if they wrongly suspect him of something.
One possible reason could be the overwhelming burden of his destiny. Constantly facing dark forces and the pressure of saving the wizarding world might have worn him down. Another reason could be the loss of his loved ones. For example, if in the fanfiction Hermione or Ron were killed, that could shatter his spirit.
One possible reason could be that in the fanfiction, Harry might have discovered some dark secrets about his parents that made him feel betrayed. For example, if he found out they had some hidden and unethical magical experiments in their past.
In some fanfictions, his family might be involved in dark magic or have a bad reputation. So, Harry Potter refuses them to uphold his own moral values. He is known for his strong sense of right and wrong, and if his family goes against that, he would distance himself.
One possible reason could be that he might feel abandoned. Growing up without them, he could resent the fact that they left him an orphan. He might think that if they were really so great, they would have found a way to survive and be there for him.
Perhaps he discovers some secrets about his parents that he doesn't like. For example, he might find out that they were involved in some dangerous or unethical magic before. This new knowledge could make him angry at them for putting themselves and their family in jeopardy.
Maybe he was tired of the constant danger and responsibility that came with magic. In the original series, he faced so many threats. In fanfiction, he might just want a normal life away from all that.
Maybe he was hit by a powerful dark magic spell specifically designed to target the soul. Such spells in the Harry Potter universe can be extremely dangerous.
In the fanfiction, it could be that Harry's aunt and uncle managed to intercept the letter before it got to Harry. They were so determined to keep Harry away from the wizarding world that they might have found a way to stop the owl or trick the postal system. They could have used some kind of Muggle - magic repelling charm they got hold of somehow. This would be a very interesting twist in the story as it shows how far they would go to keep Harry ordinary.
One possible reason could be a misunderstanding. Maybe his mate misinterprets something Harry does or says. For example, if Harry is often away on dangerous adventures, his mate might think he doesn't care about their relationship and reject him.