After watching the TFA three times, I do have some thoughts and reflections on the movie. I know there are some pretty weird ones already out there and some really dumb ones. I don't like to speculate too much, but focus on what we know and what we see.
First there is Rey. Who are her parents? We know she is strong with the Force so it is very likely that her parents too were strong with the Force too. She was left on Jakku as a young child in the hands of Unkar Plutt (that is his hand and voice in Rey's Force Flash). Whomever left Rey on Jakku wasn't interested in who's hands she was left in, only that she was alive. This would lead me to believe that she was not put on Jakku by her parents. When we see her, she mimics both Anakin's life as well as Luke's on Tatooine. Both Luke and Anakin where on desert planets way out in the middle of no where, making them hard to detect/find with the Force. This is likely the reason Rey was left on Jakku too, so she would not be found. Her abilities as a mechanic and pilot also mirror Anakin and Luke. My first impressions is that Rey is a Skywalker. However, she has no memory of who her parents are. If Luke or Han and Leia are her parents, then she has no memory left of them. I can only assume then her memory was taken from her. The names Han Solo, and Luke Skywalker are very well known to her, and the Galaxy. So more than likely they are names she would hear a lot growing up and if she knew those names to be family, she would never have forgotten them. Unless her memories where taken from her, perhaps using the Force?
Kylo Ren seems to know who Rey is. When BB-8 steals a Freighter and escapes Jakku, Lt. Mitaka tells Ren that the droid was with a girl. Ren Force pulls Mitaka to him and demands "What girl?" I think we dismiss this as Ren just being curious, but I think there is more too it. The idea that the droid escaped with help, male, female, or alien, wouldn't be of any concern to Ren. He didn't care about anyone else that helped the droid escape, just this girl. Why? Out of a universe of women, why would he be concerned about a scavenger on Jakku? Unless he knew something more. As the movie goes on and we see the interaction between Ren and Rey, this feeling that Ren knows who Rey is, only grows. So, IF Ren knows Rey, and is worried that she might be the one that Mitaka is referring too, then he knows Rey is on Jakku. If Ren knows that Rey is on Jakku, is he the one who put her there?
Han and Leia also know Rey. Maz asks "who's the girl" and we cut away. Maz senses she is Force sensitive and that there is something more too her. The cut away at this time implies that Han tells Maz who she is. Just a scavenger he picked up on Jakku, or someone more? I would lean more towards something more. Maz points to Luke's Lightsaber as something she should take. It was Anakin's and passed down to Luke, and now passed to her...another implication that the saber is being passed down to family. Maz pushes that the saber belongs to her now. Maz has to know now that she is more than just a scavenger on Jakku. What did Han tell her? Then there is Leia. After Han died and the Falcon returned to D'Qar and Leia embraced Rey, her look was one of seeing a long lost loved one meeting again. We think that she is comforting her due to the loss of Han, but this doesn't make sense to me, as the two hadn't met yet at this point and Leia would have no attachment to Rey to hug her. However, neither Leia nor Han tell Rey, "hey, you're our long lost daughter, who we thought died" so I would move away from Rey being their daughter. If she is, they are still hiding it from her to protect her from something. However, since their looks and overall feeling is relief and happy to see her, no fear in sight, I would speculate that they are not hiding her or trying to keep their daughter in the dark of them being her parents. There is no real reason too. Least none that we know of yet.
In conclusion, my theory is Rey was training alongside Ben Solo with Luke. Ben was seduced by the Darkside (or always was) by Snoke and and killed Luke's new generation of Jedi, but couldn't kill Rey, his own cousin (his Lightside struggle). Instead, Ben (now Kylo Ren), wiped her memory of her family and hid Rey on Jakku unbeknownst to everyone. Rey is more than likely a Skywalker and not Ben's twin sister. Although that is still a possibility as Rey and Ben seem to be approximately the same age. More than likely Rey is Luke's daughter, and he was so devastated at her "death" that he went into hiding. However, a even zanier twist would be that Rey is the offspring of Anakin's brother or sister. Watto did say that he married Shmi off. While we never saw her give birth to another child with her new hubby Cliegg Lars, doesn't mean that it couldn't have happened. Maybe she was preggers when she was kidnapped by the Raiders, gave birth, and the Raiders sold the baby. Or if we remove the PT, and Uncle Owen is actually a relative of Lukes/Anakins (I mean, he is dressed like the Jedi), maybe Owen and Beru had a child before Luke came into the picture and left like Biggs and Tank to fight with the rebellion.