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.
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.
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.
Well, in some fanfictions, Lizzie might go on magical adventures just like her mother. She could be exploring new spells or traveling to different magical realms. Maybe she discovers a hidden part of Hogwarts that no one knew about before.
She could be a very complex character. Physically, she might have Hermione's bushy hair but Draco's pale complexion. Emotionally, she would be a blend of her parents. She'd have Hermione's strong sense of justice and desire to help others, but also Draco's pride. This could lead to internal conflicts as she tries to balance these different aspects of her personality. For example, when faced with a bully, she might want to stand up for the victim (like Hermione) but also worry about how it would affect her own status (like Draco).
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.
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.
She could be excellent at Transfiguration, inheriting Hermione's skills in that area. Transfiguration requires a lot of concentration and knowledge, and with Hermione as her mother, she'd likely have a natural aptitude for it.
A great storyline could be the daughter having a unique magical ability that neither of her parents have. This could lead to her being both feared and admired at Hogwarts. In some fanfictions, she uses this ability to solve mysteries around the school. There could also be a story where she has to choose between following the Slytherin or Gryffindor values more closely, as she has a connection to both through her parents. This internal struggle can make for a very engaging read. And of course, a plot where she gets involved in a time - travel adventure related to her parents' Hogwarts years would be really exciting.