Harry might develop a different set of values. Flitwick has his own way of looking at the wizarding world. Under his care, Harry could grow to value knowledge and magical finesse more than he does in the original story. This could lead to a Harry who is more diplomatic and less impulsive when dealing with other wizards and witches.
Perhaps Flitwick discovers Harry has a natural aptitude for a particular type of magic that is not commonly taught. So he takes it upon himself to train Harry privately. This could lead to Harry having a special role in a future conflict, like being able to counter a dark magic spell that no one else can because of this special training.
Well, in fanfiction, Flitwick could be portrayed as having a deep understanding of ancient protective magic. He might use this knowledge to create a sort of magical perimeter around Harry. He could also enlist the help of other magical creatures or objects within Hogwarts that are known for their protective qualities. For instance, he could get the help of the house - elves, who have their own brand of magic that could be used to keep Harry out of harm's way. And he may constantly be on the lookout for any signs of danger directed at Harry, using his magical senses to detect any malicious intents.
It could strengthen Harry's relationship with the Hogwarts faculty. Since Flitwick is taking the time to train him, Harry might feel more connected to the school and its teachers. This could lead to him having more support from the faculty as a whole.
On the other hand, he could become more powerful and confident in his magic earlier. Walburga might push him hard in his magical studies, and with her knowledge of the Black family's magical heritage, Harry could develop a more assertive and perhaps even a bit of an arrogant streak in his magical abilities. But at the same time, he would also have to deal with the internal conflict of the Black family's dark past.
His allies could include Hermione and Ron as always. Since they are his best friends, they would support him in his apprenticeship. Hermione might even help him study and research with Flitwick. Neville could also be an ally as he is also interested in learning more magic. As for rivals, Draco might be jealous of Harry's new position and try to undermine him. Some of the Slytherins who don't like Harry might also be his rivals in this scenario.
Well, it would completely turn the story on its head. In the original, Snape is an adult figure and often at odds with Harry. But if Harry were to raise Snape, perhaps Snape would grow up with a different view of Harry. He might not be so bitter and resentful. Harry could instill in him values that would make Snape less likely to be drawn to the Dark Arts. This would also mean that the whole Hogwarts dynamic would change. Snape might be a much more positive influence on the students instead of the intimidating figure he was in the original story.
Harry would probably be more empathetic towards non - human magical creatures. The Mosasaurus is so different from the usual magical creatures he encounters at Hogwarts. He would understand the needs and feelings of these large and perhaps misunderstood creatures better, which could change his approach to dealing with all magical creatures in general.
One possible plot could be that Flitwick focuses on teaching Harry advanced charm spells. He might see Harry's potential and take it upon himself to train Harry privately, away from the prying eyes of Snape or other potentially hostile teachers. This could lead to Harry having a new set of skills that come in handy during battles.
One great Filius Flitwick fanfiction is 'The Charms of Flitwick'. It delves into his early days as a teacher at Hogwarts, showing his struggles and how he developed his unique teaching style. Another interesting one is 'Flitwick's Hidden Talents' which reveals some unexpected skills he has aside from charms. And there's also 'In the Company of Flitwick' that focuses on his relationships with students and fellow teachers.
One common theme is Flitwick's expertise in charms. Many fanfictions explore his skills in teaching and creating new charm spells. Another theme is his relationship with students. Some show him as a mentor, guiding students through difficult times. Also, his role in the Hogwarts faculty is often explored, like how he interacts with other professors.