In AGOT, Ned is adamant on handing over the throne to Stannis since he'd be the rightful heir despite the protests of the common opinion of the man; Lannister power over the throne and Renly wanting to be King.
Why not legitimize one of Robert's bastards and put them on the throne?
All it takes is for the king to officially legitimize a bastard so that they'll inherit and seeing as that the Hand of the King is in a lot of ways the true power behind the throne, why didn't Ned think about legitimizing one Robert's bastards and putting them on the throne? Or if he can't directly legitimize the bastard in question (Gendry for example), then just forge a letter and say that it's from the King.
Yes, the child in question would face stigma because of their baseborn status, but once the truth comes out about Joffrey then there's two choices for king: A bastard born from the king or a bastard born from twincest.
The reason I put up Gendry as my main example is because he's the oldest son and child that Ned would have most access to in order to advise and groom. Eadric Storm would also be a good choice, but he's being fostered by his uncle so there would be concerns that he's being used as a figurehead for Stannis.
It's also possible that this option would be heavily opposed by the Lannisters for obvious reasons. But I'm just curious as to whether it was a possibility for Ned, because he never even thinks of it.
It's one thing to be honorable and not to go against the norm of society, true. But, this is a case where the King dies without any trueborn heirs except for twincest-born bastards. You'd think this would be a viable exception to the rule when there's literally no one else who can take the throne apart from brothers who'd sooner destroy the realms between them than unite against a common enemy.
I also don't believe that Renly and Stannis would be too opposed since the option is being played out by their brother's best friend and the most honorable man in the Seven Kingdoms. Stannis knows Robert's seed by sight and only wanted to take the crown out of duty rather than personal greed. Renly would be initially ticked, but he could see a chance for influence by bonding with his newly legitimized nephew.
Just a thought.