Obviously, Harry Potter himself is likely to be a main character. Since it's about his encounter with a snake, he's going to be at the center of the story.
Another possible plot development is that the snake is actually a transformed wizard or witch under a curse. Harry, during the encounter, discovers this and tries to find a way to break the curse. This could involve a long and difficult journey, seeking rare magical ingredients and spells, and facing many challenges along the way. It could also bring up some moral dilemmas for Harry, like whether to trust the snake - who was once a human - completely or not.
There could be many interesting fanfictions involving Harry Potter and a snake. Maybe the snake is a magical creature with special powers that helps Harry in his adventures. For example, it could lead him to hidden secrets or protect him from dark wizards.
It could also be a snake from the Forbidden Forest. There are many unknown and mysterious creatures in there. A snake from that forest might be more in tune with the natural magic of Hogwarts. Harry, being the curious and kind - hearted person he is, might come across it during one of his adventures in the forest and start a friendship with it.
It could be Nagini. Even though in the original series she was more of a servant to Voldemort, in the fanfiction, she could have a different personality and form a bond with Harry.
Maybe it's an anonymous gift from someone who has been observing Harry. This person could be a former student of Hogwarts who has a special connection to snakes and sees that Harry has a kind heart and would take good care of the snake. So they leave the snake for Harry to find with a simple note that just says 'For Harry Potter'.
The snake's help could represent an unexpected alliance. In the Harry Potter series, snakes are often seen as symbols of Slytherin, which is usually at odds with Harry. So it's significant as it breaks the stereotype of enmity between Gryffindor (represented by Harry) and Slytherin.
The snake might use its ability to move stealthily. It could spy on the enemies for Harry Potter. For example, it can slither into hidden places where humans can't easily reach and gather important information.
Snake magic in Harry Potter fanfiction can be used to show a character's dark side. Since snakes are often associated with Slytherin and dark magic in the original series, in fanfiction, a character using snake magic might be on a path towards the dark arts. For example, they could use snake magic to control others or gain power in a more underhanded way.
She might encounter a dragon while on a quest in the Forbidden Forest. This could be a life - changing moment for her as she has to use her skills to either avoid or befriend the dragon.