What if Lyanna had given birth to a daughter instead of Jon Snow? What if Ser Arthur Dayne had fled with the newborn babe to Essos? What if Lyanna had already died by the time Ned reached the Tower of Joy? The babe, named Visenya Targaryen will grow up in Essos, Lys to be exact. She grows up under the tutelage of Ser Arthur Dayne. Ser Jorah, during his exile, meets the duo, completely by chance. Understanding who the self-exiled Kingsguard Knight is guarding, he pledges his sword and service to the young Princess. Ser Jorah knew Lyanna Stark growing up and he believes that protecting her daughter, the heir to the Iron Throne might just put him on the road to redemption. He decides to take a chance and send word to Ser Barristan Selmy about the little princess. Ser Barristan refuses the offer made by Robert Baratheon to remain in the Kingsguard and chooses exile. He sails to Lys to meet his new charge. Lord Varys, the spider, even with his little birds is kept in the dark about the princess. The three knights have pledged their life and sword to protect, defend, and train the heir to the Iron Throne. Under their guidance, the princess grows up as a commoner. All three knights have taken up common trade as sellswords in the city to keep up appearances. The princess is the exact copy of her late mother except for her eyes. They've turned into a startling purple since her birth, a notable trait of those with Valyrian blood. She also has a thin strip of Silver blond hair to the front on her right side of the head. The rest, a jet black. Ser Jorah suggests that they dye the unusual strip of her hair black as well. The eyes could be overlooked considering most of the Lysene do have certain Valyrian traits. But the hair could stand out amongst the rest and could certainly be memorable even in passing.
The Princess grows up under the watchful eyes of the three knights. The three of them know and remember what it was like to serve her grandfather, King Aerys Targaryen, aka the Mad King. Although her father, the late crown prince Rhaegar Targaryen never seemed to show any signs of the late King's madness, his actions towards Lyanna Stark was still a cautionary tale. He was married, he had children, he was next in line to sit on the Iron Throne. He was raised to be the next King of the Seven Kingdoms. But his actions during the Tourney at Harrenhal was nothing but reckless. He insulted his wife, the Starks of Winterfell, and the Northern Lords were outraged at his actions towards Lyanna and the entire kingdom of Dorne was out for blood at the treatment of their beloved princess. They would ensure that the princess would grow up knowing her history. Both of her family's histories. But how do you explain that to a child? One grandfather burned the other and laughed at the face of his pain. When would the right time be to explain to her? How would you explain it to her? But they agreed unanimously that she must be made aware of her ancestry. If she was going to be the Queen of the Seven Kingdoms, she needed to know the ugly truth of where and how she came about. She needed to learn the history of each of the Great Houses of Westeros, their allegiances, and their histories. Including both of her own. The Targaryens, with their 300-year dominance over the Seven Kingdoms. The Starks, with their 8000-year history, as Kings of Winter. The Princess, they hoped would inherit their magic, honor and, their ambitions and morals and grow up to be the Queen that Westeros needed. Robert Baratheon sat on the Iron Throne but all three knights doubted how long his reign would last. Anyone could tell he wasn't built to sit on a chair and listen to the grievances of the common people. Even with the help of his advisors, if he did manage to make his reign last, they weren't sure if it made a difference from the Mad King. At least, they hoped Robert wouldn't burn people for their actions.
The Prince Viserys and newborn Princess Daenerys were still missing, No one has any word on them since the death of Queen Rhaella. Aegon built the wheel that Westeros has been spinning on for so long. A Baratheon sits on top of it, for the moment. The three knights could only wonder what would become of the wheel when the time comes that a Targaryen sits on top again. There's no denying that the Princess is the Blood of the Dragon; they could only hope that she would also have the Heart of a Wolf
Things to note,
1) Jorah has been exiled by Lord Eddard Stark and the Princess is 3 years old when she meets him.
2) Eddard Stark becomes Lord of Winterfell and marries Catelyn Tully.
3) Jon Snow will still exist in the story and he really is Ned Stark's bastard. The mother could be Lady Ashara Dayne...that is open for your own imagination.