You can start by giving her a distinct personality. Maybe she's shy because she's always been feared for her father's sake. Then, gradually show her growing confidence as she discovers her own strengths. For example, if she has the power to raise small skeletons as minions, at first she's scared of it, but later she learns to use it for good. Also, explore her relationships with other characters. Her friendship with Percy could be based on mutual understanding of being different among the demigods.
Well, one unique plotline could be her relationship with her father, Hades. In some fanfics, she might be trying to earn his respect in a different way than a male Percy would. For example, she could use her femininity to her advantage in the Underworld politics.
She is usually depicted as a bit of an outcast among the demigods. Because of her connection to the Underworld, which is seen as a dark and spooky place, other demigods might be afraid or distrustful of her. But this also makes her a very independent and self - reliant character in fanfics.
One great fanfic could be where the Daughter of Hades has to deal with the stigma of her parentage among the other demigods. She might struggle to fit in, like trying to participate in camp activities while others are a bit wary of her powers related to the Underworld.
I really liked 'Fem Percy: Hades' Legacy'. It delved into her powers as the daughter of Hades. Since she's female, she might use her powers in a different way than the original Percy. She could be more in tune with the underworld's female spirits and use that connection to her advantage in battles and quests.
To develop her character, start with her appearance. Maybe she has dark, almost black hair like the shadows of the Underworld and pale skin. For her personality, she could be a bit of an outcast among the other demigods due to her father. This might make her a bit brooding but also determined to prove herself. Give her some quirks, like she always wears a small charm that was a gift from Hades.
One idea could be that the daughter of Hades has the power to communicate with the dead in a unique way, like seeing their memories. She could use this to help Percy solve a mystery that involves an ancient curse. Another plot could be her struggle to fit in among the other demigods because of her dark powers and the stigma associated with Hades. Maybe she discovers a hidden power within herself that is related to her father's realm and has to learn to control it while also dealing with the outside world's view of her.
In fanfiction, Percy Jackson as the champion of Hades could bring a whole new dynamic. He might have to deal with the darker aspects of the Underworld. For example, he could be in charge of ensuring the souls are properly sorted, which is a huge responsibility.
One interesting plot idea could be her struggling with the stigma associated with being Hades' daughter. For example, other demigods might be scared or distrustful of her at first. Another could be her going on a quest related to the Underworld, like trying to stop some escaped souls from causing chaos on Earth. She could also have a special power related to the dead that she has to learn to control, such as being able to see the last moments of a dead person's life, which could be both a gift and a curse as she might see some really traumatic things.
One common theme is the exploration of Hades' character. Many fanfictions try to humanize him, showing that he's not just the 'bad guy' as he's often portrayed in the original stories. For example, they might show his love for his family in a different way.