Their relationship might start off with Lucius acting more like a shadowy protector. Hermione doesn't know who is helping her at first. Once she discovers it's Lucius, she is shocked. But as time passes, Lucius starts to explain his reasons. Maybe he tells her about his regrets regarding his past actions. They could start to bond over shared interests in magic and books. Lucius might introduce Hermione to some ancient and powerful magic that she can use to protect herself further, and this could strengthen their relationship as they become more like teacher and student in some ways, while also being protector and protected.
Lucius could have a change of heart due to Hermione's bravery. When he witnesses her standing up to the dark forces fearlessly, he might be reminded of the qualities he once admired in the wizarding world before getting involved with the dark side. So he decides to protect her to preserve that bit of goodness.
Perhaps there's a plot where Lucius Malfoy is in a situation where he owes Hermione a debt. This could lead to a strange relationship where he has to follow her around and protect her. As time passes, he starts to develop unexpected feelings for her.
You can try searching on fanfiction websites such as Archive of Our Own (AO3). It has a vast collection of fanfictions from various fandoms, and there's a good chance you'll find some stories related to Lucius and Hermione there.
There are also some fanfictions that explore an alternate universe where Lucius and Hermione meet under different circumstances, like in a magical school where there are no Voldemort - related threats. In these stories, their relationship can develop in a more positive and less dark way, and you can often find them by looking for AU (alternate universe) tags on fanfiction platforms.
Hermione would face a great deal of prejudice from the pure - blood society. Since she is Muggle - born, many in the wizarding world, especially those in the Malfoy's social circle, would look down on her. She would also have to deal with Draco's own internalized prejudices. He has been taught to think of Muggle - borns as inferior, and Hermione would have to work hard to change his view. Another challenge would be maintaining her own identity. The Malfoy family has certain expectations and a way of life, and Hermione would have to find a way to be true to herself while also fitting into this new family dynamic.
You can try fanfiction websites like Fanfiction.net or Archive of Our Own. Just search for 'Hermione and Lucius Malfoy' on their search bars, and you should be able to find relevant fanfictions.
One possible plotline could be that Lucius has a change of heart after seeing Hermione's bravery. Maybe in a dangerous situation at Hogwarts, he realizes that she is crucial to the wizarding world's future and decides to secretly protect her from dark forces that are after her knowledge or power. Another could be that Lucius is under some sort of spell or curse that makes him see Hermione as someone he must protect, like a strange form of redemption arc for him.
If Lucius wants Draco to marry Hermione, it could lead to a lot of political implications within the wizarding world. The Malfoys are a powerful pure - blood family, and Hermione's friendship with Harry Potter and association with Dumbledore's side could be seen as a way for Lucius to gain some sort of advantage. For Draco, this could be a way out of the dark path he was on in the original story. He could become more of an ally to Hermione and her friends. And Hermione might find herself in a very strange position of having to balance her own values with the expectations of the Malfoy family.
Another plot is a sort of forbidden love story. Given their different allegiances in the original series, a fanfiction might explore how Hermione and Lucius overcome their differences and fall in love. It could involve a lot of tension, secrecy, and internal conflict as they deal with their feelings and the reactions of the wizarding community.