Pre-Series changes
Jon Arryn’s daughter, who gets called Daella, makes it even if Jeyne doesn’t so both she & Elia marry Rhaegar, uniting the Falcon with the Dragon & the Sunspear. Elia still has her two while Daella has a boy named Ronnel. Daella helps hatch the dragon eggs, with one each for each of them since she has Targaryen blood due to her namesake while one does bond with Elia while Viserys gets one from elsewhere and can be one of three eggs; purple & gold egg from Aegon II, Aegon V’s white egg with green swirls or his namesake’s undescribed egg.
The black one gets named Balerion while Viserion does become the name of the pale cream one that bonds with Elia and the green one gets called Shrykos Jade. Possible names, for Viserys’ dragon, are Taryen (purple & gold) royal looking so named after their House, Springbreeze due to the green looking like leaves floating on a breeze or any other name of an ancestor or past dragon.
Or Stannis Baratheon finds a girl in Essos, who has Baratheon & Lannister ancestry with blonde-streaked black hair, one eye green & the other violet or maybe blue eyes with purple & green rings in them. This girl is a sorceress from one of the Northern Free Cities who knew to come to meet up with Stannis.
Or an unwanted daughter of an important House of Volantis gets kidnapped at 10 by a couple of sorcerers who use blood from a shadowcat & bone fragments from dragons to change her & upon her & Rhaegar’s 13th year, they dream of each other. They prepare for when Stannis goes to Volantis so he can pick her up & bring her to him with the dragon eggs that she collected plus the one dragon that hatched for her. These eggs didn’t include the seven from Summerhill but includes the eggs from the series (Drogon’s, Rhaegal’s, & Viserion’s eggs). She names herself Inanna meaning "Lady of the Heavens" as that is what she knows her parents wanted to call her.
Or Brandon Stark has a twin who is a Warg (Skinchanger) who does the impossible as a Griffin locates her and bonds with her also a shadowcat, while she also bonds with a regular wolf, a sea eagle, a kestrel, a sparrowhawk, and a viper.
Another idea that popped into my head: Inhuman female connected to Rhaegar like Dany & Harry in "Dragons United" who at 15 gotten bitten by an experiment of a cross between a Komodo Dragon & a Bat i.e. someone trying to make a dragon during World War II so her mother hit her with the Terrigen Crystals to save her life, therefore she become an ice dragon-shifter; ice powers from her Terrigenesis.