One way could be in his relationships. He might have made new friends or mended broken relationships. For instance, if he had a falling - out with Ron before the summer, during the summer months in the fanfiction, he could have worked on getting their friendship back on track.
She might have become more confident in her magic skills. In some fanfictions, the summer could be a time for her to practice independently and master new spells, thus growing in confidence when it comes to using magic.
Emotionally, he could have changed a great deal. If in the fanfiction he faced some personal challenges over the summer, like dealing with the loss of a friend or a new family situation. This might have made him more withdrawn or on the contrary, more determined to face his enemies. For example, if he discovered some secrets about his family that made him question his identity, it would have a profound impact on his emotional state.
Often, he becomes more self - reliant. Since he can't always count on others during the summer away from Hogwarts, he learns to solve problems on his own. For instance, if there's a magical creature causing trouble near the Dursleys' house, he has to figure out how to deal with it without the help of teachers or friends right away.
In some fanfictions, Harry might change physically. He could have grown taller or more muscular from any summer training he did. For example, if he was secretly practicing Quidditch on his own, it could have affected his physique. Also, mentally he could have become more mature. Maybe he spent the summer reading advanced magic books and this broadened his understanding of magic, making him more confident and less impulsive in his actions.
Well, physically he could change too. Fanfic writers might describe him as growing taller or stronger during the summer break. Emotionally, he might deal with his past traumas better. For example, if he had nightmares about Voldemort, he could learn to overcome them, perhaps through some new magical techniques he discovers or with the help of his friends. This could make him more resilient and less haunted by his past, which is a significant change for his character.
Hermione would be a likely helper. She's always into books and knowledge, so she'd probably be right there with Harry, sharing her insights and helping him find relevant information.
Well, one possibility is that Snape could start to see Harry in a new light. Maybe he discovers Harry's hidden talents or interests during the summer. For example, Harry might show an aptitude for potions that Snape had previously overlooked.
There's also 'Summer of Transformation for Harry'. Here, Harry's summer is filled with self - discovery. He starts to train his body and mind more rigorously. This leads to him becoming more confident and assertive when he returns to Hogwarts. He also makes new friends outside of the wizarding world during the summer which broadens his view of different cultures.