His character could develop in terms of his relationships. He might start to value loyalty and honesty more, especially in his friendships and marriages. This could lead to him making different choices in who he surrounds himself with. For example, he may choose to have more true friends rather than just political allies.
Perhaps their relationship begins with curiosity. Daenerys has heard of the powerful Robert in Westeros, and Robert is intrigued by this Targaryen with a claim to the Iron Throne. As they communicate more, they find common ground in their desire for a just Westeros. They might start to fall in love slowly as they share their dreams and fears, and work towards building a better future for the Seven Kingdoms.
Perhaps a plot point could involve his relationships. If he is reborn, he may handle his marriage to Cersei Lannister differently. He could try to build a more genuine relationship with her or find a way to avoid the animosity that was there before. This would have a huge impact on the political and family dynamics in Westeros.
They might interact with a father - daughter - like dynamic. Robert could be strict but also show moments of tenderness towards Myrcella. He may scold her when she makes mistakes but also praise her achievements.
He would be more aggressive. As a Baratheon, he'd inherit that family's bold and sometimes brash nature. Instead of the somewhat reserved Harry we know, he'd be more likely to take immediate action, whether in a duel or in dealing with a problem at Hogwarts.
Good fanfiction about Robert Baratheon also needs to capture the essence of the world he lives in. It should include the political intrigue, the power struggles, and the cultural aspects of Westeros. If it can tie in his actions as king with the greater political landscape, like how his decisions affected the different houses, it makes for a great read. Also, delving into his vices, like his drinking problem, and how it impacts his rule and those around him is a plus.
One idea could be a story where Robert Baratheon discovers Jon Snow's true parentage earlier. This could lead to a complex relationship where he sees Jon as a potential heir or a powerful ally. He might take Jon under his wing, train him in the ways of leadership and warfare, much like he did with Ned Stark's other children.
Power is a common theme. Fanfictions often explore how Robert wields his power as king, whether he uses it wisely or misuses it. Another theme is love and lust, like his relationship with Cersei and how that affected his rule.
If Ned Stark sees the future, he might become more distant from Robert Baratheon. Knowing what's to come, he could see Robert's actions as leading to disaster. So he might start to oppose Robert more openly, which would strain their friendship.
To develop a male OC Baratheon, start with his physical appearance. Maybe he has the typical Baratheon black hair and blue eyes, but with a scar on his face from a battle. Then think about his personality. He could be proud like most Baratheons, but also have a sense of humor.
In Reborn fanfiction, Fem Naruto's character often develops by facing new challenges. For example, she might have to adapt to different social norms in the Reborn world, which forces her to grow as a person.