There would be chaos. Joffrey is a king, so his death would create a power vacuum. His mother Cersei would be furious and would likely start a manhunt for Arya. The Lannisters might declare war on Arya's family or allies. The political landscape would be completely disrupted as different factions would try to take advantage of the situation.
In the fanfic, Arya might kill Joffrey during a big event. She could blend in with the chaos. Since she has great agility and combat skills, she could quickly approach Joffrey while everyone is distracted. For example, during a tournament or a grand feast. She could jump from a balcony or some hidden place, stab him swiftly, and then disappear into the crowd before anyone realizes what has happened.
Sansa Stark is frequently in these fanfictions. As someone who was a victim of Joffrey's cruelty, if he is in a time loop, Sansa might find herself in a different situation each time. Maybe in one loop Joffrey starts to treat her better, which would completely change her perspective and her own actions within the story. Also, Cersei Lannister could be involved. As Joffrey's mother, she would be very concerned about any changes in her son during the time loop and might try to interfere to keep things as she wants them.
Power struggles are a common theme. Just like in the original Game of Thrones, fanfictions often explore how different characters, including Ros, navigate the power dynamics in Westeros. Survival is another key theme. Characters have to find ways to survive in a world full of danger and political intrigue.
Yes, there might be some good ones out there. You can start by looking on fanfiction websites. Some may feature interesting storylines with these characters, perhaps exploring their relationships or adventures in unique fictional settings.
In some fanfiction, it might be Varys. He was known for his intelligence network and his ability to work from the shadows. He could have seen Joffrey as a threat to the stability of the Seven Kingdoms. So, he arranged for the young king's death, perhaps thinking that a more malleable ruler could take his place and be more easily influenced to make better decisions for the realm.
In fanfic, the relationship between Joffrey and Sansa is often a complex and twisted one. Joffrey was cruel in the original story, but in some fanfics, writers might explore a different side of him in relation to Sansa. It could be an exploration of what might have been if Joffrey had not been so evil, perhaps delving into a more romantic or at least a more nuanced connection between them.
In a fanfiction, the lemon could play a magical role. Imagine Joffrey is cursed, and the only way to break the curse is through a lemon - related task. This could lead to a whole new side of Joffrey being explored as he has to rely on something as simple as a lemon to save himself.
In fanfiction, a smart Joffrey often has strategic thinking. He's able to plan ahead and manipulate situations to his advantage, much like a chess player. For example, he might outwit his enemies in court politics.