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 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.
She might have an adventure where she has to compete against Beauxbatons students in a magical tournament. Hermione would use her quick thinking and vast knowledge of magic to hold her own. She could learn some new and elegant spells from the Beauxbatons students during the competition.
One idea could be that Hermione is raised in secrecy by someone who knew Voldemort's true identity but didn't want her to follow his dark path. She discovers her true parentage through a series of mysterious events, like finding an old diary of Voldemort's that reveals secrets. Then she has to struggle with her own identity, whether to embrace the part of her that is related to Voldemort or completely reject it and continue to fight for good.
She could have adventures that combine her magical skills with Clint's combat abilities. For instance, they could go on a mission to stop a villain who is using both magic and advanced technology. Hermione's spells would come in handy, while Clint's archery skills would provide a different kind of offense.
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.
In the fanfic, the LaTeX hood might give Hermione the ability to communicate with magical creatures in a unique way. For example, she could use LaTeX - like codes to understand the complex languages of dragons or centaurs. This would make her a key figure in the wizarding world's attempts to co - exist with these creatures.
In such fanfictions, we might see Hermione having a different kind of connection to knowledge. Since Athena is the goddess of wisdom, Hermione could have an even deeper understanding of magic and its history. For example, she might be able to decipher ancient spells more easily or have insights into the origins of magic that others don't. This could also change her relationships with her friends. She might become more of a leader, guiding Harry and Ron with her enhanced wisdom.
Another aspect to expect is a new set of relationships. Maybe she'll have some sort of connection to Greek demigods or other mythological creatures. This could lead to some exciting crossovers in the story. For instance, she could meet a Greek demigod who helps her with a particularly difficult magical task.
She might face a lot of internal conflict. On one hand, she has been raised with the values of Gryffindor, like courage and justice. But on the other hand, she has this dark heritage. This could lead to a really interesting character arc where she has to constantly battle her inner demons. For example, she might be tempted by the power that her relation to Voldemort could potentially bring, but she has to resist it to stay true to herself.
She could have adventures with magical farm animals. Maybe there are unicorns in the fields that she befriends and they help her protect the farm from dark magic threats.