Some much talked about ideas and how I feel about them.
SPOILERS for STAR WARS AHEAD
Apparently nothing is set in stone on who Rey's family is because Abrams wanted to leave up some possibilities for the creative teams in the next movies. Rey was waiting for her family to return which means she was: hidden to protect her, her family (or at least part of it) was killed or they really did abandoned her.
Rey is Anakin Skywalker Reborn?: I don't like this idea and there really isn't any solid proof for this. Even the line Maz has about seeing the same eyes in people wasn't directed at Rey but rather Finn. Yes Anakin was unique given his "there was no father" status but I think people are forgetting ROTS. Palpatines' talk with Anakin has him heavily hinting that either his master or he created life (Anakin) through the force. You could argue their a tool of the force but he wasn't just a creation that happened to spring up. This theory also reduces Rey by making her someone else when we already explored Anakins' story.
The granddaughter of Obi-Wan Kenobi?: As much as I'd love to have a Kenobi legacy I can't see this happening. There's no real evidence to support this besides the fact she learned mind tricks fast, her accent and a couple other details that really aren't that concrete. Rey used s lot of Jedi maneuvers and only had an "awakening" after Kylo Ren tried to read her mind. All Jedi can learn these things, Luke used it too and the only reason it didn't work on Jabba was because he wasn't weak minded. As for her accent I recall it being referred to as a "core" accent. Many politicians used it (look at Padme when she's playing the queen) and many picked it up living on the planet. Sure it's possible Rey picked it up from a parent but she could have also picked it up from anyone.
Rey is a Solo?: One of the two best theories although it's still a little flawed. If Han suspected he never says anything although the visual Dictionary for the movie seems to imply her name being from a helmet she picked up. He wouldn't recognize the name Rey if it was his daughter and he's not force sensitive. Leia and Rey comforted each other although they don't know each other prior to this. Why didn't Leia just hug Chewie instead? Maybe Leia sensed she was upset or something else. Remember when she found out the truth about her own family? There was shock but Leia stated that somehow she always knew. If Rey was her kid I'd think she'd have more of a reaction even if she has to keep it a secret (such as the need to have Rey confront Ren.) Finding a long lost kid after losing another to the dark side would warrant more than a simple embrace.
Rey is a Skywalker?: This one has the most evidence to support it. A lot of this can also work with the Solo theory since Leia is also from the Skywalker family tree but these traits are what we define as Skywalker traits. Rey excels at mechanics, is a great pilot and has a connection to the Anakin/Luke lightsaber. The way she dresses is similar to the way Anakin and Luke dressed in ep 1 and 4 which is also similar to the Jedi style. The music cue when she finally takes the lightsaber is Lukes' theme. She has a close relationship with a droid, is offered a job by Han and gets annoyed with someone overlooking her as a pilot. Of course this also raises the question why Leia doesn't say anything but it's easier to explain. It's not Leia's place to say it's Lukes' and sending Rey to him is the perfect time to do so. Both of them have emotional looks on their faces when they meet. It's been suggested that Rey suddenly knowing how to use the force is her remembering her training.
Whether she's a Solo or a Skywalker raises the question of why she was left on another planet. It is strange that Rey happens to be on the same planet as Lor San Tekka, the man with the remains of the map to find Luke that supposed to be a family friend. He's the one that keeps bringing up Kylo Rens' family. There's also the fact Kylo Ren seems to know who she is to the point he gets enraged at the mere mention of a girl. Since he's still conflicted at this point he'd likely feel the same around the time the Knights of Ren slaughtered the new Jedi academy. Maybe he saved Rey because he couldn't have her around or kill her?
If Rey is either it means that Leia should know their related. It's iffy how much Han suspected although Maz specifically asked him who she was. Given Hans' long time relationship with Maz it's unknown how much she'd tell him. I can't see Han acting so cool and collected if this was his lost kid if he knew. If Luke is her father then he might have even more baggage to deal with than just training a student. There would be emotional distress if she was his niece too given what happened to his nephew but a daughter would cut deeper. Losing his students and his nephew going to the dark side is bad enough but losing his wife/lover and a kid? The guilt of not being their for a child after what he went through with his own dad? That would be brutal. I'm not sure if he'd be ready to go over everything again.