A Harry Potter/Game of Thrones/Eragon crossover idea I had. I would write it myself, but I don't have the skill with words to do it justice.
Eragon and Saphira appear in Westeros on the Isle of Faces a significant amount of time after leaving Alagaesia. There they meet Rheagar Targaryen and Lyanna Stark. Eragon befriends Rheagar, then decides to explore Westeros and Essos. As such, he is not present during Robert's Rebellion. He decides to watch over what remains of Rheagar's family from the shadow's. When Daenerys is married to Khal Drogo, he reveals himself to her and becomes her sworn sheild.
A 14 year old Harry Potter and Daphne Greengrass appear in Tyrion's chambers during his time as Hand of the King. The only change from Harry Potter canon is that Harry and Daphne started dating in 4th year. As such, Daphne was his date to the Yule Ball and his hostage in the 2nd task. After appearing in Tyrion's chambers, Harry becomes Tyrion's squire and Daphne becomes Sansa's handmaid. Whatever method used in getting them from Hogwarts to Westeros needs to destroy the horcrux in Harry's scar, leaving Harry with Voldemort's magical knowlege.
Eragon places Rider spells on dragon eggs
Dany becomes the Unburnt through an assassination attempt by Viserys that kills Khal Drogo and Viserys
Jon learns of his parentage before Ned goes south and instead of joining the Night's Watch, he goes across the Narrow Sea to join what remains of his family
Daphne is not Slytherin's Ice Queen, but has a cold exterior to those she doesn't know or like
One egg hatches for Dany, and one for Jon. The third can hatch for the character of your choice