One possibility is that she discovers a hidden potion recipe in her father's old notes. She decides to try and make it, facing various challenges along the way, like finding rare ingredients and dealing with unexpected reactions. Maybe she uses her knowledge to help out her friends at Hogwarts who are in trouble.
The plot could center around Snape's daughter entering a potions competition. She has high hopes but also a lot of pressure because she's Snape's daughter. Along the way, she might struggle with some potions that require a delicate touch. She could also get help from an unexpected source, like a friendly ghost who has some ancient knowledge about potions.
If she is shy, she might prefer to work alone on her potions. For example, she would spend hours in a quiet corner of the potions dungeon, carefully measuring and mixing ingredients without much interaction. On the other hand, if she is outgoing, she could be more likely to collaborate with other students, sharing ideas and perhaps even creating new potions through group efforts.
Her character could be developed as a bit of an outcast at first. People might be intimidated by her being Snape's daughter and assume she's as strict and unapproachable as he is. But as she shows her love for potions, she could gradually make friends. For example, she might start sharing her knowledge about potions with other students who are struggling, which shows her kind side beneath the initially aloof exterior.
Well, Draco could be punished by Snape for being disrespectful in Potions class. Snape might order Draco to write a long essay on the history and significance of a particular potion. This would not only be a humiliation for Draco but also a chance for him to gain more knowledge about potions. And it would add an interesting dynamic to their relationship as Draco would be resentful at first but might end up learning something valuable.
Snape could also punish Draco by deducting house points. Since Draco is in Slytherin, losing house points is a big blow to his pride and to the whole Slytherin house. This would make Draco the target of his housemates' ire, adding to the punishment's severity.
In some fanfics, Snape may use a potion similar to the Felix Felicis, the luck potion. But instead of just bringing luck, this modified version could be used to change Harry's fate or the course of his actions. It could make Harry more inclined to make certain choices that Snape desires. Also, there could be a potion that can temporarily suppress Harry's magic, which Snape might use to keep Harry out of trouble or to protect him in a different way.
I really like 'The Unforeseen Consequences of Snape's Potions Mishap'. In this fanfiction, Snape's accident in the potions lab creates a magical cloud that spreads throughout Hogwarts. It starts to make everyone act out of character. For example, the Gryffindors become overly studious and the Slytherins start being overly friendly. It's a really fun take on how a potions accident can have widespread and unexpected effects on the Hogwarts community.
Snape is likely to be initially suspicious. He has long held a rather negative view of Harry, so when he discovers Harry is good at potions, he might think Harry has some ulterior motives or has received improper help. After all, in the regular Hogwarts setting, their relationship is quite tense.
If she discovers some of Snape's old potions recipes or magical knowledge, she could share it with the Hogwarts faculty and students. This would not only improve the potions classes but also make her a respected figure in the school. And her presence might also make the students re - evaluate their views on Snape, which would in turn change the overall atmosphere at Hogwarts.