It could be portrayed as a fierce and powerful protector. Always by Percy's side, ready to attack any threat that comes their way.
Perhaps the wolf is a magical being in the fanfiction. It has a shiny coat that glows in the moonlight, and its eyes hold a mysterious power. It could be a symbol of freedom and wildness, and Percy's relationship with it could represent his connection to the untamed aspects of nature.
The wolf might be shown as a very intelligent creature. For example, it could understand complex commands from Percy and even figure out solutions to problems they encounter during their adventures. It may also have a gentle side, especially when it's just with Percy, like a big softie at heart.
Often, Aphrodite's charmspeak in fanfiction involving Percy is portrayed as a kind of irresistible force. It can make Percy act out of character, perhaps being more flirtatious or compliant than he usually is.
In some fanfictions, Percy's possessiveness of Nico is portrayed through his words. He could be very vocal about how Nico should stay close to him or how he doesn't like others being around Nico. Also, his body language might play a part, like always having a hand on Nico's shoulder or standing very close to him to mark his territory, so to speak.
Well, in such a fanfiction, Percy as the pet wolf would likely be involved in Artemis' hunts. He could have unique interactions with the other members of Artemis' group. His human side might conflict with his wolf instincts at times. And there could be a whole story arc about how he got turned into a wolf and his journey to either get back to his human form or fully embrace his new identity as Artemis' pet.
In the fanfiction, perhaps Percy found the wolf as a cub, abandoned in the wild. He decided to take it in and raise it. As the wolf grows, it becomes his loyal companion. They might face challenges from people who fear the wolf because of its wild nature, but Percy stands up for his pet.
One way to portray Percy's family differently in fanfiction is to give them more power. For example, Sally could be a powerful seer like Rachel. This would change how Percy interacts with his mom as she could foresee some of his future battles and give him advice based on her visions. Another way is to make Paul more of a fighter. He could have some hidden combat skills and help Percy in his battles against monsters.
Often, Hermione's intelligence is pitted against Bellatrix's madness. The fanfiction might show Hermione trying to outwit Bellatrix while Bellatrix uses her dark magic to keep Hermione under her thumb, like a pet in a way. For example, Bellatrix could be using some dark spell to control Hermione's actions, and Hermione has to use her wits to break free.
They are often portrayed as more powerful. Writers like to make Percy's lightning powers stronger to create more intense battle scenes. For example, he can send bolts of lightning that can split mountains.
Well, in some fanfictions, Percy's fire powers are a secret at first. He has to hide them from his friends and enemies alike. This secrecy can create a lot of tension in the story. As he tries to figure out where these powers came from and how to use them properly, he might face challenges from other demigods who sense this new and strange power within him.
In fanfic, Percy's pain tolerance might be demonstrated through his ability to keep fighting even when severely injured. For example, he could have a broken bone or a deep cut, but still be able to face his enemies. This is because he has trained himself over time, through all his adventures. He knows that giving in to pain could mean endangering his friends, so he pushes through it, which is a common theme in many fan-written stories about him.
It could be a hyena. The Joker has been associated with hyenas in the past, and it would make sense for him to have a new, more special hyena as his pet.