I recently returned to the alternate-history discussion forum after months of self-ban to avoid distractions from my new career and my writing projects. Since I want to finish Little People, Big Guns (finish, revise/edit, and submit to an interested publisher) by the end of the summer, I'm self-banned again, but for awhile I was reading, commenting, and making sure my newsletter sign-up link is in my post signature.
(Seriously, that's a good idea if you're a writer.)
One thing I found (and spent most of a day reading) is a story called "Were We Different Men." I posted the fanfiction.net link so you can read it without having to join the site. It diverges from the canon timeline during the execution of Lord Rickard Stark, father of Brandon Stark (who in canon died alongside his father), Ned Stark (the protagonist of the first book), Lyanna Stark (who was abducted or eloped with Prince Rhaegar Targaryen), and Benjen Stark.
In the books Rickard demanded trial by combat and Aerys the Mad King decreed that fire would be the champion of House Targaryen. So Rickard was hung from the ceiling in his armor over a fire that roasted him alive. Brandon was put in a noose with a sword just out of reach and strangled himself trying to rescue his father. In this story, the noose breaks when Brandon is near-death from strangulation. He's got a hellacious scar around his neck and his voice is damaged, but he lives. Aerys forces him to watch his father roast alive and then dumps him in the Black Cells--where he's rescued by Ser Barristan Selmy, the eunuch Varys, and young Jaime Lannister, who then flee the capital.
As a result, Tywin Lannister is not restrained by having his eldest (and favored) son Jaime as Aerys' de facto hostage and joins Robert's Rebellion immediately. The Reach is forced to participate more actively (as opposed to just sitting there besieging Stannis in Storm's End) and although Robert's Rebellion is shorter, it is bloodier.
Here are some highlights of the story:
*Quellon Greyjoy, who sought to integrate the Iron Islands into the wider Westerosi world and rejected "the Old Way," lives much longer than in canon. This has some interesting consequences when some of his ideas lead to Westeros trying to compete with the Iron Bank and other banking institutions in the Free Cities and to suppress piracy and slaving in the Stepstones...
*Lysa ends up much better off than her resented elder sister Catelyn (who was slated to marry Brandon before his canonical death) in the marriage department. This is a positive situation for Lysa, not a negative one for Catelyn--Brandon's experience in King's Landing sours him on the out of control behavior that led to him getting the reputation as "the wild wolf." We have a happier and saner Lysa this time around.
*Barbrey Dustin has a happier life than in canon.
*Robert is a much, much better king than the lazy whoremonger he was in canon. Not going to go into detail for spoiler reasons, but he gets a much more intimate encounter with how unbelievably awful Aerys is and vows to do better.
*Cersei Lannister gets religion and becomes a much, much, MUCH better person than she was in canon too.
Of course, it wouldn't be George RR Martin if things didn't go wrong. The "interesting consequences" are near-apocalyptic in scope for Westeros and the Others are stirring again. The story is so well-regarded it even has a TVTropes page, which you can see here.