A friend of mine and myself speculated about this for a LONG time one day. Too long.
Here is what might have happened:
Rey was training as a Jedi under Luke when she was very very young (which is why she has some Jedi powers that were easily accessible). Kylo Ren was also training under Luke. They are brother and sister. So…both children of Han and Leia.
When Kylo turned against Luke and killed the Jedi that we’re in training, he wanted to kill Rey, but found himself unable to either because they are siblings or because she was so young. Almost like Luke was unable to kill Vader because of the family connection. So instead of killing her, he took her as far away from everyone as he could get her within the galaxy hoping that she would just be forgotten. Jakku is known to be very remote and distant from everything. He either sold her or gave her away to Unkar Plutt (who you can see in Rey’s vision and hear him say “Quiet girl!). It’s definitely him, so there is no denying that.
Thinking that Rey was dead (which is likely not her real name and would also account for why they didn’t recognize the name ‘Rey’), Han and Leia moved on with their lives and eventually split up (as some parents do when they are blaming each other for a child’s death or some other tragedy). I’m thinking along the lines of “The Missing” here. This would account for Han Solo not knowing who Rey is but some of the looks that he gives her also probably make him think, “I wonder if my daughter would have been like her?” It would also account for Leia not knowing who she is, but she might have known through the Force who she was as soon as she saw her. The leaked script said that Leia gives Rey a “mother’s embrace”.
When Kylo Ren discovers who she is and that “a girl” (like when he says “What Girl?!?”) is helping the droid and Finn, and after Rey begins to use the Force, he makes the comment that she is going to become stronger with every passing moment or something to that effect. He knows exactly who she is since he was responsible for her disappearance and she only reappeared after the incident on Jakku.
Rey is likely not a Kenobi. The timeline doesn’t make sense. Obi-wan never had a relationship with anyone in any of the movies and is much too serious of a Jedi for a relationship, which is against the Jedi code. Marriage is what began the downfall of Anakin. Plus the movie writers would have to invent a serious side story creating the possibility for too many plot holes that don’t connect to the other movies at all.
Rey is likely not Luke’s daughter because that seems much too obvious and easy for the story. It wouldn’t be much of a plot twist or really interesting at all. Plus Luke is a very serious Jedi like Obi-wan was so it’s doubtful that he had a relationship at all. I know in the canon books he had a relationship with Mara-Jade, but it’s obvious they have abandoned most of the canon stories. Luke would have had to have a relationship and a kid sometime within the 5–10 years after Return of the Jedi to produce Rey. Not to mention the look that he gives her at the end of the movie wasn’t a “Hello daughter!” look. It was a “so you’re the young Jedi hopeful that has put all the pieces together and found me” look. Luke was waiting for someone with some kind of Force strengths to come and find him. R2-D2 probably was only told to turn on when someone with the Force comes looking for him.
So there you have it. Rey is the long lost daughter of Han and Leia (which makes her part of the Skywalker linage), and the sister of Kylo Ren. Brother vs Sister.