Well, an interesting plot could be the OC son being sent on a diplomatic mission. He has to use his wits and the Lannister name to gain the upper hand. He could also be in a situation where he discovers a long - hidden secret about the Lannister family that changes his perspective on everything. Maybe he has to decide whether to keep the secret or expose it for the greater good.
One idea could be the OC's struggle to live up to his father's reputation. He might face challenges within the Lannister family, like dealing with his siblings' envy or rivalry. Another could be his role in the political games of Westeros, perhaps trying to form alliances different from his father's.
One idea could be the daughter getting involved in the political intrigues of King's Landing. Maybe she has to navigate the complex relationships between the different houses while trying to uphold the Lannister name. Another could be her having a secret talent, like being an excellent strategist, which she uses to help her father in unexpected ways.
One idea could be a political alliance. Rhaella, being a Targaryen, and Tywin, a powerful Lannister, could form an uneasy alliance to deal with a common threat to the Seven Kingdoms. They might start off distrustful of each other but gradually come to respect one another's political acumen.
One plot idea could be that Tywin, seeing Sansa's value as a political pawn, decides to use her in a different way than originally planned. Instead of marrying her off to Tyrion, he takes her under his wing to teach her the ways of politics in Westeros. This could lead to an unlikely bond between them as Sansa tries to navigate the dangerous world of the Lannisters while still holding onto her Stark identity.
One plot idea could be Arya, in disguise, getting close to Tywin as a servant in King's Landing. She could overhear his plans and use that knowledge to try and undermine the Lannisters from within. Another could be a scenario where Tywin discovers Arya's true identity but decides to keep her alive for some strategic reason, perhaps to use her as a bargaining chip against the Starks. And a third idea might be a story set in an alternate universe where Arya and Tywin form an unlikely alliance against a common, greater threat, like a White Walker invasion in the south before the events at the Wall.
The novelty of original characters. They can bring new perspectives to the world of Tywin Lannister. For example, an OC could be from a different social class and show us how Tywin interacts with those he normally wouldn't deal with.
To develop the character, consider his backstory. What was his childhood like? Did he have a special tutor or mentor? His experiences outside the Lannister stronghold are also crucial. If he travels and meets different people, it can change his views. For instance, if he meets a group of poor peasants and sees their struggle, it might make him more empathetic. Also, his moral compass can be different from his father's. He could be more inclined towards justice and fairness, which would lead to conflicts within the family.
One plot idea could be an OC who is a healer and tries to cure Cersei of a mysterious illness that no one else can figure out. This could lead to a complex relationship between them as Cersei is not used to depending on others.
Often, there's a plotline where the OC is a servant or a low - born character who rises to prominence and has to deal with the politics of Tywin Lannister and the Starks. They might be caught in a web of lies and intrigue as they try to serve both sides without getting killed. Also, a plotline could involve an OC as a mage or a seer who foresees the destruction that the feud between the two houses will bring and tries to stop it.
In such fanfiction, the relationship can be complex. Tywin Lannister is a complex character. An OC (Original Character) might be drawn to his power and authority at first. But his cold and calculating nature could also create conflict. For example, the OC might try to humanize him, which would be a difficult task considering his reputation.
Well, perhaps in this fanfiction, Harry Potter could end up in Westeros and meet Tywin Lannister. Harry, with his knowledge of magic, could be seen as a threat or an asset by Tywin. Tywin might try to use Harry's magic for his own military purposes, like using magic to enhance his army's weapons or defenses. Harry, on the other hand, would have to navigate the cut - throat politics of Westeros while trying to keep his magic a secret from those who might misuse it.