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.
Maybe in the fan fiction, Tywin discovers Arya's true identity but decides not to kill her immediately. Instead, he tries to use her as a pawn against the Starks. He might try to train her in the Lannister ways, which would put Arya in a difficult position of having to resist while also trying to find a way to outwit Tywin.
Their relationship could develop in a very complex way. At first, Arya would be full of rage towards Tywin. However, if she were to be captured by the Lannisters and Tywin were to show her a different side of himself, perhaps a more paternal side (even if it's just a ploy), Arya might start to question her hatred. On the other hand, Tywin might see Arya as a potential asset, someone he could mold in his own image. This could lead to a relationship that is part mentor - mentee and part captor - captive, with a lot of power dynamics and emotional turmoil.
Well, in fan fiction, Arya might not be as afraid of Tywin as she potentially was in the original series. Since fan fiction allows for more creative freedom, she could have some special powers or skills that give her an edge over Tywin. Their interaction could then be more of an equal battle of wits and wills, rather than the one - sided power dynamic that was somewhat implied in the original.
In OC fanfiction involving Tywin Lannister and the Starks, Tywin's strategic mind is a central aspect of his portrayal. He is always looking for ways to strengthen his house's position. The Starks, on the other hand, are known for their integrity. An OC could be a healer, for instance, who treats the wounded from both sides during a battle. This OC's presence might lead to some unexpected interactions between Tywin and the Starks, like Tywin seeing the Starks in a more sympathetic light due to the OC's influence.
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.
Often, Tywin is in a position of power over Sansa. So, he may interact with her in a condescending or manipulative way. For instance, he might use his authority to force her into certain decisions.
Well, in some fanfictions, Sansa might learn from Tywin about power and leadership. Since Tywin is a master of political intrigue, Sansa could start to pick up on his strategies. For example, she could use his teachings to gain more influence in the court, while also trying to keep her Stark identity intact. This would create an interesting dynamic between them as she tries to navigate the dangerous world of the Lannisters.
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.
Arya and Tywin could also interact in a more unexpected way. Maybe in a situation where they are forced to work together for a short time due to some external threat. During this time, Arya has to constantly remind herself of her ultimate goal of revenge while Tywin is constantly on guard against her, yet also seeing potential in her skills for his own purposes.