Her character might develop in a way that she constantly tries to prove herself. Because of her parents' polar - opposite reputations, she would feel the need to show that she is not defined by their actions or reputations. She could excel in both academics and magic - related sports, like Quidditch, to gain respect.
Her character could develop through her relationships. She might befriend students from all houses, which is a bit unusual in Hogwarts. This would be because she has qualities from both Gryffindor (from Hermione) and Slytherin (from Snape). Through these friendships, she'd learn different perspectives and grow as a person. For instance, she could teach her Gryffindor friends about the subtleties of Slytherin thinking and vice versa.
Her character might develop through her magical education. As she learns more spells and potions, she also gains wisdom. She could start to question the actions of her Lestrange ancestors and try to make amends for their wrongdoings. Maybe she uses her mother's knowledge of ancient magic to find a way to heal the wounds left by the wizarding wars.
She would likely be more confident in her abilities. Since Hera is a powerful goddess, Hermione as her daughter would have a sense of divine entitlement. She might be less hesitant to take charge in situations.
Hermione's character would likely experience a great deal of conflict. On one hand, she has her own strong values and beliefs which are in direct opposition to the Malfoy family's pure - blood ideology. But on the other hand, she might be forced to adapt to her new living situation. She could become more guarded and learn to play the political games within the wizarding society as she tries to survive in the Malfoy household.
Their relationship might start with a lot of hostility. Hermione, with her strong sense of justice, would be very wary of Lucius, who has a history of being on the wrong side. But as they interact more, Lucius could be intrigued by Hermione's intelligence and courage. He might start to change his ways to impress her, and Hermione could see the potential for good in him.
Their relationship could develop in a rather complex way. Given their very different backgrounds, there would be a lot of clashes. However, the marriage law forces them to coexist. As time passes, they might find common ground in their love for the wizarding world, albeit from different perspectives. Hermione could introduce Lucius to the idea of a more inclusive wizarding society, and Lucius could teach Hermione about the long - standing pure - blood traditions that have some value, if reformed. This mutual exchange could lead to a grudging respect and eventually something more like friendship or even love.
Hermione's character might develop a new sense of understanding towards those on the 'dark side'. Through her time - traveling experiences with Lucius, she could see that not all who seem bad are completely irredeemable. She may become more open - minded and less quick to judge based on a person's family or initial appearance.
Well, it might start off very coldly. Lucius would be reluctant to accept help from a mudblood like Hermione in his eyes. But Hermione, being persistent, would keep pushing to help him. As she starts to break down his walls, Lucius might find himself drawn to her in a strange way. Maybe he starts to see her as a powerful ally rather than an inferior. And Hermione might start to understand the pressures that made Lucius the way he was before. This could lead to a relationship that has elements of mutual understanding, respect, and perhaps even a bit of attraction as they both discover new sides of each other during their time together.
For Lucius, his development could be a slow process. At first, he resents being around Hermione, but as he witnesses her magic and her kind heart, he starts to question his own values. He may begin to see the error of his ways regarding blood purity. Hermione might develop more patience. Dealing with Lucius, who has such different views, makes her learn to tolerate and try to change someone rather than just oppose them.
She could be a Ravenclaw. Given Hermione's love for knowledge and Draco's cleverness, their daughter would likely have a thirst for learning and a sharp intellect. Ravenclaw values wisdom and intelligence, which she would surely possess in abundance.