Maybe the plot could revolve around Lizzie's magical abilities. She could inherit unique magical skills from both Hermione and Draco. For example, she might have Hermione's brilliant academic approach to magic but also Draco's natural aptitude for some darker arts (not in an evil way, but in a more complex understanding). This could lead to her getting into trouble with the school authorities as they misunderstand her intentions. And she has to prove that she's using her abilities for good, perhaps by solving a mystery that threatens Hogwarts.
One possible plot could be that their daughter is a young witch growing up in a world that still has remnants of the old wizarding world rivalries. She could struggle with being the daughter of two former enemies, facing prejudice from some in the wizarding community. She might have extraordinary magical abilities that draw both positive and negative attention, and she has to learn to navigate her own identity while also trying to bridge the gap between the Gryffindor and Slytherin legacies her parents represent.
She could be very determined. If she sets her mind on something, whether it's winning the House Cup for her house or solving a mystery in the Forbidden Forest, she won't give up easily. Her determination could come from Hermione's never - say - die attitude and Draco's single - mindedness when he wants to achieve something.
Well, considering her parents, she might have a bit of a rebellious streak. Since Draco was from a traditional pure - blood family and Hermione was a Muggle - born witch, their daughter could be defiant against the old - fashioned pure - blood ideologies. She could be very outspoken and not afraid to challenge the status quo in the wizarding world. And she'd likely be very empathetic too, having seen different sides of the wizarding community through her parents.
Well, she could be highly intelligent like Hermione. Maybe she'd be a bookworm, always with her nose in a tome, eager to learn new spells and knowledge. At the same time, she might inherit Draco's sense of pride. So she'd be confident in her abilities and not easily intimidated. However, she could also have a kind heart, a blend of Hermione's compassion and Draco's loyalty to his own, which in this case would be his family.
She could be highly intelligent like both her parents. Since Hermione is known for her studiousness and Draco is quite cunning in his own way, their daughter might inherit a sharp mind. She could be very confident, perhaps a bit of a loner at times as she might feel she doesn't quite fit into the typical Hogwarts houses' stereotypes. And she might have a strong sense of justice, but also a bit of a mischievous streak from Draco.
One possible plot could be that Draco's daughter is starting her first year at Hogwarts. She's nervous about living up to the Malfoy name. She might make friends with some unexpected people, like a Muggle - born student, which causes some tension in the family as Draco has to come to terms with his daughter's more inclusive views.
Well, it could involve Lyra growing up in the Malfoy family. Maybe she has to deal with the family's reputation. She might struggle to make friends outside of the pure - blood circle her father Draco was part of, and she could be trying to find her own identity in a world that has certain expectations of her because of her family name.
She might have a great relationship with her mother, Hermione. Hermione would surely pass on her wisdom and knowledge to Lizzie. Lizzie could look up to her mother and seek her advice often.
In a fanfiction, they could be living together in a secret location. Hermione is on a mission to research a very rare and dangerous form of magic, and Draco has some knowledge about it from his family's library. They form an uneasy alliance. As they live together, they find that they have to rely on each other more than they thought. Draco helps Hermione with the more Slytherin - like aspects of their research, like secrecy and cunning, while Hermione teaches Draco about the importance of kindness and justice. Their living situation becomes a catalyst for personal growth and a potential romantic relationship.
One possible plot could be that Draco, for some reason, becomes Hermione's servant. Maybe it's part of a magical pact or a punishment from the wizarding world. Draco, being used to his pure - blood superiority, would initially resist but gradually learn to respect Hermione's intelligence and kindness. They could then go on adventures together, facing various magical threats and learning more about each other's true natures.