It could be a snowy owl, just like Hedwig. In the original series, Hedwig was Harry's loyal and beautiful owl. A fanfiction might follow this idea and have Harry find a young snowy owl that he bonds with immediately.
The pet might introduce Harry to new magical creatures or areas. For instance, if it's a cat that loves to explore the Forbidden Forest, it could lead Harry to discover hidden places or meet other magical beings that he wouldn't have encountered otherwise. This could expand his knowledge of the wizarding world and his own powers.
In fanfiction, Harry Potter could have a phoenix as a pet. A phoenix is a very magical and majestic creature. It can be a great companion for Harry. It has the ability to heal itself by bursting into flames and being reborn from the ashes, which could be useful in many dangerous situations Harry might face.
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 plot could be that the cat is a magical key to a hidden chamber. As a child, Harry stumbles upon the cat and realizes it has a strange mark. When he touches the mark, it reveals a map that leads him to a secret chamber filled with ancient magic and knowledge.
There's no way to know for sure without more details. He might be a cat. Cats are often used as pets in fanfiction too. They can be independent and mysterious, which can add an interesting dynamic to the story.
Well, it could be about a character who is forced to hide their true identity and pretends to be a pet in the wizarding world. Maybe they are on the run from some dark wizards and find a safe haven by disguising as a pet in a Hogwarts dormitory.
Perhaps the hellhound has a special power like being able to travel through fire unharmed. This power could play a crucial role in the story. For example, the main character and their friends get trapped in a burning building, and the hellhound uses its power to create a path through the fire, allowing them to escape. The fanfiction could then focus on how they keep this power a secret from the school authorities.
In some Harry Potter fanfictions, Voldemort's pet could be a large and menacing snake, like Nagini. Since snakes are often associated with Slytherin and Voldemort was a descendant of Slytherin, it makes sense for his pet to be a snake. It could also be some sort of dark magical creature, perhaps a basilisk under his control. Another possibility is a magical raven or crow, which could be used as spies for him, flying around Hogwarts and other places to gather information.
One possible plot could be that a new, mysterious pet appears in the pet shop in Diagon Alley. This pet has some unique magical abilities that no one has ever seen before. A young witch or wizard discovers it and tries to figure out its origin and how to take care of it. Maybe it's a key to an ancient magic mystery.
Perhaps Draco could be a Kneazle. Kneazles are intelligent and independent, which would suit Draco's personality. As a Kneazle, he might be constantly sneaking around and getting into places he's not supposed to. Harry would have to use his wits to keep Draco out of trouble and also earn his trust, as Kneazles can be quite picky about who they bond with.