One of the main themes is family dynamics. The Lannisters have a complex family structure, and the OC has to find his place within it. Also, the theme of survival is prevalent. In the cut - throat world of Game of Thrones, the OC has to fight to stay alive and relevant. And then there's the theme of reputation, as the Lannister name comes with both advantages and burdens.
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.
One great 'OC Son of Lannister' fanfic is 'The Lannister's Heir'. It explores the journey of the OC as he tries to navigate the complex politics of Westeros. Another is 'A Lannister's Redemption' which shows how the character battles with the family's reputation.
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.
If the male OC Lannister was a proud and ambitious type, he would likely interact with Cersei with a mix of respect and rivalry. He'd try to gain her favor while also making sure she didn't see him as a threat. With Tyrion, he might initially look down on him but later come to respect his intelligence. And when it comes to Jon Snow, he could start off as dismissive but as he realizes Jon's leadership skills, he might try to form an alliance.
If Naruto was a Lannister in a Game of Thrones fanfiction, he'd be a very different kind of Lannister. He'd probably use his ninja skills to protect the family in ways they've never seen before. Instead of the usual backstabbing, he might bring in a sense of loyalty and honor that the Lannisters lack. He could use his jutsu to make the Lannister stronghold more secure, like creating barriers with his earth - style jutsu.
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.
There could be a fanfiction that focuses on an alternate universe where Tywin survives longer. In this story, he would have to deal with the changing political landscape after the events at King's Landing. He might try to regain control of the situation, perhaps by making alliances with unexpected parties. His character would be forced to adapt to new threats and challenges, which could lead to some very interesting plot developments.
One great 'Game of Thrones Tywin Lannister Fanfiction' is 'The Lion's Legacy'. It delves deep into Tywin's past, showing his rise to power in a very detailed and engaging way. The author does an excellent job of portraying his complex personality.
I really liked 'Jaime Lannister: A Different Path'. In this fanfiction, Jaime decides to side with the Starks early on. It's an interesting take as it completely changes the course of his character in a believable way. There are some great character interactions that are not in the original show.