One interesting plotline could be Snape finally seeing Harry's potential in potions and secretly mentoring him outside of class. This could lead to a grudging respect between them.
Snape's character might develop by becoming more patient. Usually strict and harsh, teaching Harry could force him to be more patient and understanding, as Harry has a different learning style.
One key moment could be when Snape first explains the concept of Occlumency to Harry. He might describe it as a mental defense against mind - reading and intrusion. For example, Snape could say it's like building a wall around your thoughts. Another moment could be when Harry starts to practice and has his first small success in blocking Snape's attempts to penetrate his mind. This might give him a sense of confidence and also make Snape see that Harry has potential in this area. A third key moment could be when Harry uses Occlumency in a real - life dangerous situation, like when facing a powerful dark wizard who tries to read his thoughts to find out some crucial information.
At first, it strains their relationship. Snape is still bitter towards Harry because of his father, and Harry doesn't trust Snape. But as the lessons progress, they start to understand each other a bit more. Harry realizes that Snape has some valuable knowledge to offer, and Snape sees that Harry is trying hard. This slowly builds a grudging respect between them.
The potion could enhance Harry's magical powers. For example, it might make his spells stronger or more accurate. This would be useful in battles against dark wizards or in his general studies at Hogwarts.
Dumbledore can also be a significant character. He might be the one who arranges for Snape to teach Harry, having some grand plan in mind. He could be observing from the sidelines, guiding Snape's actions and Harry's progress subtly.
I'm not sure if there is a specific fanfic precisely on this combination. There are a vast number of fanfics in the Harry Potter fandom, and while there could be one out there that involves Snape, Hermione, and a potion related to a baby in some way, it would take some extensive searching through fanfic platforms to find it.
One interesting plot could be that Snape, while teaching Harry, discovers a hidden talent in him for a very rare branch of magic. He then decides to secretly mentor Harry in that area, leading to a unique bond between them.
Well, perhaps in the fanfiction, a villain uses a love potion on Harry to make him fall for Snape in order to create chaos in Hogwarts. Snape then has to deal with Harry's sudden affection while also trying to uncover the plot behind it. They might go through a journey of self - discovery and realizing that there are deeper emotions hidden beneath the false ones caused by the potion.
Snape's strict teaching style makes Harry more resilient. Harry has to endure Snape's sarcasm and difficult tasks, which in turn helps him grow as a wizard. He learns to face challenges and not be intimidated by authority figures in the magical world.