Even before viewing Star Wars: The Force Awakens I was damn sure that Rey would be the title character of the film and that she was a Skywalker. After another viewing of the film with another coming up I'm going to go over points in the movie which prove that she's a part of this legendary family. Spoilers to follow
Point 1: She dresses exactly like every other member of the Skywalker family and has a similar desire to be a pilot. This is shown by her playfully putting on a rebel helmet and looking up the the sky.
Point 2: She has the same back story as every other Skywalker give or take a detail. This of course being the tale of getting abandoned by family on a remote desert planet in hopes that she would be hidden from the dark side. Same as Luke in "A New Hope" and a slight change on Anakin being born along with discovered on a desert planet. They all dream about leaving, but are stuck with their families until something forces them to leave. (The force forces them).
Point 3: When her memory is being extracted by Kylo Ren she imagines a beauty island surrounded by water, then at the end when she meets Luke it's the exact place being described. Didn't notice this bit of information until I re-watched the film.
Point 4: She has a strong sensitivity with the force despite no training or knowledge. That's also where the title come from as the Force is being awakened within her. Only the Skywalker family has such a high level of power and grasp of the force.
Point 5: It makes sense that Luke would want to separate himself from her and not have her trained since the family has always had problems with being tempted by the dark side. Out of love he probably created a space where she wouldn't have to deal with the sorrow of having your family destroy the balance of the force constantly.
Point 6: The force clearly calls upon Rey during the scene where she touches upon Anakin's lightsaber. It's a lightsaber that has always been a central part of the story and in the hands of the central family. Both have felt connected to it and also had their hands cut off. So, hopefully that doesn't also happen to Rey!
Point 7: Apparently Kylo Ren calls her cousin in the Disney Infinity 3 playset though I can't confirm that. Update: False via Kotaku post states: Curses!
Point 8: (new) After a third watching it seems that Han somewhat knows who Rey is after looking to take care of her with a job and letting Maz know during their talk when Rey discovers the lighsaber. Maz says the family isn't returning, "but, there's someone who still could" to which Rey replies "Luke". Even further throughout it seems that Leia also had an idea of who she was comforting her after Han's death instead of Chewbacca which is odd considering how close they would have been.
I'll be watching the film plenty more and will add to the list when I discover new things. Just from the evidence collected I would be shocked if she wasn't a Skywalker. Also, the actress Daisy Ridley did a hell of a job with the character. Strong emotion and amazement really set her aside from the other actors. Again, in case that wasn't clear this is speculation based on what I've seen and looked up online. We really won't know anything for sure until Episode VIII or maybe even Episode IX.