Well, some fanfictions might focus on Ariana's magic. Since her magic was unstable, a story could revolve around her learning to control it in a different way than what Albus and Aberforth thought was best. This could lead to a split in their family not just because of the incident with the Muggle boys but also due to differences in how to handle Ariana's powers. Or, there could be a time - travel based fanfic where a character from the future tries to save Ariana from her tragic fate, changing the course of the Dumbledore family history.
A really interesting plot could be that Ariana's spirit is trapped in a limbo of sorts. Harry, with his penchant for getting into extraordinary situations, discovers a way to communicate with her. They then work together to free her spirit. Along the way, Harry learns more about Dumbledore's past through Ariana's memories, which in turn helps him in his own battles against the remaining dark forces in the wizarding world.
Perhaps Harry discovers a hidden room in Hogwarts that was Ariana's. As he explores it, he starts to feel a strange bond with her. He then uses magic to try and bring out her memories or essence. As he gets to know more about her, he starts to see her as a friend and someone he wants to protect, much like he does with his own friends.
The lap kiss might symbolize a deep emotional connection or a turning point in the story. It could be a way to show a rare moment of vulnerability or affection for Ariana Dumbledore.
In this fanfiction, Ariana is an AI. She could be a unique creation within the fan - made world. Maybe she was created by a brilliant wizard who was inspired by the real Ariana Dumbledore. But she is not a human, rather a technological or magically - created intelligence that has a role to play in Harry's life.
Well, Harry might outsmart Dumbledore by using his unique understanding of the magical world in ways Dumbledore doesn't expect. For example, Harry has a more direct experience with some dark magic items due to his encounters with Voldemort. He could use this knowledge to make decisions that seem counter - intuitive to Dumbledore but are actually more effective in a particular situation.
He might give Harry detention. Harry could be made to clean the trophies in the trophy room or help Hagrid with some difficult tasks in the Forbidden Forest under strict supervision.
One possible reason could be that Harry breaks school rules in the fanfiction. For example, if he ventures into the Forbidden Forest without permission. Dumbledore is strict about the safety of students and such disobedience might lead to punishment.
For Harry, his friends like Ron and Hermione are likely to be his allies. They've always been by his side through thick and thin in the original series, so it's natural for them to support Harry in a confrontation with Dumbledore. Neville could also be an ally, given his growth and courage shown in the books.
Well, it might be a fanfic where a new dark wizard emerges with a unique power to manipulate memories, and they target Dumbledore's memories of Harry Potter. Or it could be a story where Dumbledore gets caught in a complex web of magical experiments gone awry, leading to this memory loss.