SPOILERS IF YOU ARE NOT "UP-TO-DATE" ON STAR WARS LORE.
If "Skywalker" is no longer just a name, but a title, Rey wont be the main architect or founder of the new order. This supposes that the Skywalker order is based purely on maintaining the balance of the Force, no more taking sides of light vs. dark. (Back to the origin of Legends stories) so here's a list of official canon names of those who helped build the new order.
1) Qui-gon Jin. First of the known Jedi to begin the trials of the Whills. The Foce "priestess" who over saw Jin in his trials. Jin did not complete his trials, so never became a proper Force Ghost. But could speak to Force users if they listened. (*Clone Wars cartoon, Revenge of the Sith)
2) Yoda. At the urging of Jins spectorial voice completes his trials, in essence the first proto-Skywalker. (*Clone Wars cartoon, Revenge of the Sith)
3) Obi-wan Kenobi. Presumably completed his trials while in exile (*Revenge of the Sith)
4) Ahsoka Tano. The apprentice to Anakin you never saw in the movies. Served with Skywalker just after Attack of the Clones to just prior to Revenge of the Sith. Left the Jedi order and traveled around as more a Gray Jedi. Years later shows up again as a more Balanced looking Force user. (*Clone Wars cartoon, Rebels.)
5) Kanan Jarrus (former Padawan Caleb Dume) Learned the ways of Balance from a Entity known as "The Bendu" while still believing himself to be a Jedi, was probably on the road to becoming a Skywalker. (*Rebels)
6) Bendu. Possibly a physical representation of the balance in the force, and the origin of the original "jedi" order. Quote: "Jedi and Sith wield the Ashla and Bogan. The light and the dark. I'm the one in the middle. The Bendu." (*Rebels)
There are other representations of aspects of the Force known as "the Mortis Gods."
The Father, possibly the cosmic Force.
The Daughter, the light, possibly the Living Force. And The Son, The Dark. (*Clone Wars, Rebels)
Now. With all that, I propose that Rey will be like Luke in the Return of the Jedi, they didn't start the Jedi Order, wasn't really a member of the old order, but a new start. A start on the beginning the Skywalkers.
