I'm not sure specifically as there could be many interpretations. It might be about Ned giving something (could be an object, a secret, or a responsibility) to Jon Stark in a fan - created fictional story.
Yes, there might be. The Game of Thrones fandom is vast, and many creative fanfics exist. You can start by looking on popular fanfic websites like Archive of Our Own or FanFiction.net.
Perhaps the OC son has a special ability or gift that Jon Arryn was trying to protect him from being exploited for. After Jon Arryn's death, the son has to learn to control this ability while also dealing with the power struggles in King's Landing. He could be a key figure in the fight against the White Walkers, using his unique skills to help the living, all while trying to uphold his father's legacy.
Well, he could be a character who has been reimagined. Jon Arryn in the original series was an important figure. In this fanfic, the 'Brightflame' addition might suggest some sort of connection to a more flamboyant or perhaps magically - inclined lineage. It's possible that he has abilities or a backstory that ties him to this 'Brightflame' concept, which could make him a key player in the fanfic's plot, different from his role in the original story.
The concept of heir in a 'Game of Thrones' fanfic involving Jon Arryn and Brightflame is highly variable. There are so many factors at play. The world of 'Game of Thrones' is full of complex family trees and political relationships. It's possible that an heir could emerge from a previously unthought - of branch of the family. Maybe there's a secret marriage in the past that ties a character to both Jon Arryn and Brightflame in a way that gives them a legitimate claim to be the heir. And then there are the power - hungry lords and ladies who might try to manipulate the situation to their advantage, putting forward their own candidates as the heir, whether they have a real claim or not.
The 'Game of Thrones' world is vast and full of possibilities in fanfiction. Regarding Jon Arryn's wife in a Blackfyre - related fanfic, it's likely that the fanfic author has crafted a unique story. It could be that the wife is a character who is a part of a secret Blackfyre plot, or she might be a noblewoman from a house that has some connection to the Blackfyre legacy. However, without actually delving into the fanfic, we can only speculate about who she might be.
When writing a 'jon arryn oc son fanfiction', it's important to research the world in which the story is set. Know the politics, the culture, and the other characters. For example, you should understand the power dynamics in the area where Jon Arryn held influence. You can then build your plot around how the OC son fits into this world. Maybe he wants to continue his father's legacy or break free from it. Also, don't forget to add some conflict, like an enemy or a difficult situation that the OC son has to face.
One popular 'jon arryn oc son fanfiction' might be the one where the OC son is on a quest to uncover the secrets of his father's past. It often has elements of mystery and adventure.