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.
Well, in some fanfictions, he might find it in the Forbidden Forest during one of his adventures. Maybe it was injured and he felt a connection to it, so he decided to take it as a pet.
Perhaps he would name it 'Slyther', as a play on the Slytherin house name. Given that snakes are often associated with the Slytherin house in the Harry Potter universe, it would be a fitting name for his pet snake.
Obviously, Harry Potter is one of the main characters. And the snake, which could be a unique creation within the fanfiction, is also a main character. It might have its own personality and abilities that are different from normal snakes in the Harry Potter universe.