Power is a very common theme. The Lannisters are known for their lust for power in the original series, and fanfictions often explore how they gain, maintain, or lose power. Family loyalty is another. Despite their many flaws, the Lannisters have a strong sense of family loyalty, and this is often a central theme in fanfictions.
In the Game of Thrones novels, Tyrion Lannister remains alive. His character undergoes many challenges and developments but doesn't meet his end. The author has kept him as an important part of the ongoing narrative.
Consider their upbringing. If they were raised away from the main Lannister stronghold, they might have a different set of values. Maybe they were raised by a kind - hearted servant who taught them to be more empathetic. So their personality could be a mix of Lannister pride and this new - found empathy. They could be conflicted between loyalty to their family and doing what is right according to their own conscience.
Perhaps an OC Lannister who is a talented healer. In a world full of wars and battles in Game of Thrones, a healer Lannister could be an interesting addition. They might have to deal with the moral dilemma of healing both their own family's enemies and allies. They could also have a secret love affair with a member of another house, which would add a layer of drama.
There are no appropriate'sex stories' in a positive sense about Cersei Lannister. Her character is more about her ambition and her role in the power struggles of the Seven Kingdoms. Her relationships, like with Jaime which was incestuous, were part of her downfall and her complex character arc. It was a way to show the moral decay in the world of 'Game of Thrones' rather than just a titillating story.
One possible plot could be that Hermione, with her intelligence and magical skills from the Harry Potter universe, is transported to the world of Game of Thrones. She meets Jamie there. Maybe she tries to use her knowledge to prevent some of the bloodshed in Westeros. She could team up with Jamie, who has his own complex history and skills with a sword. Together they might try to bring about a more peaceful solution among the warring houses.
Well, 'Lannister in Exile' is a good one. It tells the story of a Lannister who has been exiled from Westeros and must find a way to survive and regain his family's honor in a foreign land. Another interesting fanfiction is 'The Lannister Heir'. This one focuses on an unexpected heir in the Lannister line and the power struggles that ensue within the family as a result.
One possible plot could be a young Lannister male OC who, unlike his power - hungry relatives, is more interested in the arts and knowledge. He could secretly befriend a maester and learn about ancient Valyrian lore. This would put him at odds with his family as they see his pursuits as a waste of time. He might then use his knowledge to try and broker peace during the war, facing many challenges along the way.
Redemption is also a big theme. Jamie has a lot of baggage in his original story, and in crossovers, his journey of self - discovery and trying to make up for his past mistakes can be explored further in different settings. For example, in a crossover with a more morally strict universe, he might strive even harder to redeem himself.