One possible fanfiction could start with Robert Baratheon as a young man. He is full of ambition and dreams of a united Westeros under his rule. He trains hard in combat, his sword becoming an extension of himself. He meets his future friends, like Ned Stark, during these formative years. They share tales around the campfire and make grand plans for the future. As he grows older, he participates in battles, his victories making him more and more renowned. His charisma draws people to him, and soon he is seen as a natural leader. But with power also comes temptation, and he has to face the challenges of politics and intrigue that come his way.
One interesting fanfiction could be one where Robert Baratheon survives and has to face the challenges of ruling Westeros in a more peaceful way. He might have to deal with political intrigues without the use of brute force as much. Another could be a story about his youth, exploring his relationship with Ned Stark before all the chaos of the Iron Throne began. And there could also be a fanfic that focuses on his relationship with his children, had he been more involved in their upbringing.
You can start by looking on fanfiction websites like Archive of Our Own (AO3). It has a large collection of 'Game of Thrones' fanfictions, and you can specifically search for ones related to Robert Baratheon. Another option is FanFiction.net. There are many writers on there who might have created interesting stories about him.
One idea could be a fanfiction where Robert Baratheon's son, perhaps Joffrey (even though he was a cruel character), has a change of heart. In this fanfic, he could start to see the error of his ways after meeting a wise old maester who tells him tales of great but just kings. He then tries to make amends for his past actions, which would be a huge departure from his original character arc in the show and books.
To make a fanfiction about Robert Baratheon interesting, you can give him new challenges. Maybe he has to deal with a magical threat that the original series didn't show him facing. You could also explore his backstory more. What was his life like before he became king? His upbringing, his first loves, and his early battles could all be interesting areas to explore. Additionally, changing the power dynamics around him can be great. For instance, if some of the minor houses suddenly become more powerful and he has to negotiate with them. This can lead to all sorts of interesting political intrigue and character development.
She could also be shown as a great warrior in certain fanfictions. Despite the norms of the society in Game of Thrones that often restrict women's roles in combat, she breaks free from those constraints. She trains in secret and then emerges as a surprise defender of her family when they are under attack.
Another aspect that fanfictions often explore is her relationship with other members of the Baratheon family. In some stories, she could be at odds with Cersei, seeing through her false charm and manipulative ways. This creates a power struggle within the family, which is a common and engaging theme in Game of Thrones fanfiction.
If the fanfiction makes Robert Baratheon's son a master of disguise, it would add a new twist. He could infiltrate the different houses and spy on their plans. He might discover plots that were otherwise hidden, like a secret alliance between the Lannisters and the Greyjoys to take over the Iron Throne. His ability to move undetected would give him an edge in the game of thrones and also create a lot of suspense as the other characters are constantly being watched without their knowledge.
One interesting fanfiction could be one where Renly Baratheon survives and forms a different alliance. Maybe he teams up with the Starks against the Lannisters, which would completely change the power dynamics in Westeros.
Love and loyalty also feature prominently. An OC Baratheon might be torn between loyalty to their family and their love for someone from another house. This creates a lot of internal conflict and interesting plotlines. In some stories, an OC Baratheon's loyalty to a friend or a romantic partner can change the course of events in the political landscape of Westeros.