Alright so my theory is that Rey is not related in any way to the Skywalkers/Organa-Solo families but that she is related to the group who used to protect the first Jedi temple and was on the island that Luke now inhabits.
"It is time then." Luke said sighing as he turned towards Rey on the cliff. He had felt the changes in the force and knew someone would be coming for him. He had felt the death of his friend at the hands of his own son, Ben, the one who called himself Kylo Ren now; Luke's former student. He had felt the conflict in Ben for days leading up to the confrontation. He had felt Rey's anger and pain as she watched Han die.
He had felt her use that anger against Ben in the forest, and hoped that her early training would win over and he wouldn't lose another child to the dark side. And when he felt her balance her anger, calm her mind, he had felt hope for the first time in a long time.
But now she was here on the island, the island that she couldn't possibly remember and the thought made Luke sad. He frowned at Rey as he walked towards her.
"You have come to learn the ways of the Jedi?" He asked.
"I have come to figure out how I am doing these amazing things that I have no previous knowledge of." She said frowning. "I am not entirely sure what a Jedi is or does but you have answers to the questions I have. If those lead to the path of a Jedi, so be it." She squared her shoulders fully prepared to be sent away. Leia had explained Luke's choice to disappear after Kylo Ren had slaughtered all of the Jedis-in-training that fearful rainy night.
"Spoken like a true Ren." He smiled sadly at Rey and felt what he had feared so long ago.
"What is that supposed to mean?" She asked flabbergasted.
"It means that you are honest. It means that I will teach you as much as I feel I can teach you. But there is a history in you, in your blood that you do not yet know." He folded his hands inside of his robe and moved to take the steps down to the village. "And it will take time. Concentration, commitment."
"Whatever it takes." She said rushing after him.
"Do not commit so easily to something you know so little about." He said stopping and turning back towards her.
"I know that I feel something and it scares me. Sometimes it flows through me like a trickle. Sometimes like a raging waterfall but always there. I know that there are few who can stop the oncoming darkness and yet I feel as if I am the only one who can stop it." She said becoming embarrassed at expressing the way she felt in words. It sounded brash and young and foolhardy. "I want to control it, instead of it controlling me."
"Controlling the force isn't possible." Luke said turning again to continue his descent.
"Then what is it that Jedi's do if not control the force?" She asked bracing herself on the rock wall as she tried to keep pace with the Jedi before her.
"We bend it to our will. We use it but always remember that it is larger than us. That it flows through us and uses us for what we seek out in the universe. If you seek darkness, it will use you for darkness." He said stopping to look out over the ocean again. "If you seek light, it will use you for light. But do not be mistaken into believing that you can control it."
Rey felt Luke's sadness coming off of him in waves. It was clear how heavily Kylo Ren weighed on his mind, even after all these years.
"So how do I find balance?" She asked staring at the man who might become her reluctant mentor. Luke looked at her then and smiled.
"That is for you to discover. I am a Jedi, I am not the balance but the light side. My darkness has cooled to a dull ache. But yours, I can feel the conflict in you. Just as I could with Ben. You have both the light and the dark."
"I think I understand." She said quietly, hoping that she did.
"Not yet you don't." He said calmly, "but you will."