What if Hoster let Lysa keep Petyr's baby?
If Hoster decided to let Lysa keep the pregnancy, Petyr would not have been shipped off but instead, remained at Riverrun to recover. He would have forced them to marry and things would have been cold between them as Petyr did ot love Lysa and began to hate her once he realized that it was her and not Catelyn who he ad been with.
Due to Lysa's young age, her pregnancy was difficult and she ended up dying a few days after her daughter was born from childbed fever. The last thing she did was name her daughter, Myna Baelish.
Hoster, not wanting Petyr and a reminder of his now dead daughter at Riverrun, gives Petyr the sum of Lysa's dowery and sends him and Myna to the Fingers. Robert's Rebellion still goes on and plays out normally. Petyr makes contact with Catelyn (who writes to keep tabs on her niece) and eventually this is how Jon Arryn gives Petyr the post in Gulltown. Like in the books and show, Petyr rises high and becomes Master of Coin, taking Myna to King's Landing to continue raising her there.
Things at KL are a bit different. Petyr becomes rather close with Jon Arryn for the sole purpose of getting close to Jon's heir, Harold Hardying. A betrothal between Myna and Harry is drafted and once it becomes set in stone, Jon Arryn suspiciously dies. However, this time Petyr is not behind it, as he had an invested interest in Jon Arryn living until at least the wedding.
Ned and the rest come to King's Landing and Petyr is taken aback by Sansa. Myna takes notice of this and crafts a plan. Things play out similarly in the first book, with the exceptions of Myna and Sansa becoming close friends. When Ned is executed, the war breaks out and Petyr stays "loyal" to the crown. He still goes around the kingdoms sowing chaos, while slowly trying to keep Harry out of the war. Myna is the one caring for Sansa when she is being beaten and bruised by Joffrey.
Petyr is still given titles and prestige, even though he kind of would have had that through his short marriage to Lysa and the money from her dowry.
Red Wedding still happens, leaving Myna and Petyr Sansa's only remaining family.
Sansa still ends up marrying Tyrion but only because she told Myna of the plan for her to marry Wilas Tyrell, something neither Myna nor Petyr want. This forces them to temporarily leave Sansa behind when they head to the Vale. However, instead of Petyr marrying Lysa as he does in the books, they are going for Myna's wedding to Harry, the crown wanting to bring the Vale to their side.
Purple wedding happens and Sansa is whisked away, the same way she was in the books. Petyr poses her as his bastard daughter Alayne, something Myna reinforces upon her marriage to Harry. Everyone believes Myna and Alayne to be sisters, since they look similar when Sansa dyes her hair.
Myna becomes with child and is slowly convinces Harry to move onto Winterfell, to take it back from the Boltons. She tells him who Sansa really is and states that she is family, or at least, she hopes it to be. Petyr, slowly begins seducing Sansa during their time at the Vale. Previously, most of her interactions where through Myna, and when she finally gets to know Petyr, she trusts him a bit more easily than she did in the books/show.
Slowly, she falls in love with him and marries him about six months after Myna gives birth to a young boy. Harry leads an army in the name of his wife's family to reclaim Winterfell. He succeeds but dies in the process. Myna isn't exactly heartbroken as she didn't love Harry and has already given him a son, making her Lady of the Vale and Protector of the Vale until he reached maturity.
Sansa, the last Stark, moves into Winterfell with her now husband, Petyr. They have several children, as they wait out the chaos that happens in KL and surviving the Long Night.
