This is the most popular theory about Rey's origins. It's popular because there is an overwhelming amount of evidence for it. The most important of which is: just look at how they stare at each other in the last moments of The Force Awakens! That's all the evidence you need.
Alright, you need more? Rey's story mirrors Luke's. Although she grew up an uncultured orphan on a desert planet, she emerges as a natural pilot and lightsaber-er. And when Rey touches Luke's lightsaber, she has a force vision, showing that she is strongly linked to its previous owner.
Also, Kylo Ren -- if Luke is Rey's father, that makes Ben her cousin -- seems to be aware of her existence, showing interest when someone mentions "that girl" on Jakku. Ben must know he had a young female cousin that disappeared after he murdered Luke's padawans.
Finally, Luke's lines in the trailer (not included in the movie) support this theory. He says, "The Force is strong in my family. My father has it. I have it. My sister has it. You have that power, too." While the trailer doesn't show who he's speaking to, we can infer from context clues. As he says these lines, he touches R2D2.
The only person who interacts with both Luke and R2D2 in Episode VII is Rey. In fact, R2D2 gives Rey the information she needs to reunite with Luke. Because he's outlining how the Force is passed down from generation to generation, we can assume Luke is talking to his child. Put all these details together and it really looks like Rey is Luke's child.
It makes sense that after his nephew turned to the Dark Side, Luke would want to protect his daughter by hiding her. The Skywalkers aren't great on originality. By Luke's generation, you'd think the Skywalkers would know to avoid sand! It's like the Averys and the Manitowoc County. Just move away already!
This is a slight variant on the popular fan theory above. Perhaps Luke hides his force-sensitive daughter on Jakku not to protect her from everyone but to protect everyone from her.
Compare what Rey is wearing to what the Jedi students in Episode III are wearing. Beige FOR DAYS. Redditor -caribou- points out that Rey's outfit resembles a worn and torn Jedi uniform.
This suggests that Luke had begun training his daughter when she was very young because DAMN the force is strong in that one.
After seeing how his nephew used his powers to murder a bunch of people, Luke might have feared Rey's great potential could turn her to the Dark Side and decided that he needed to keep her as far away from the Jedi life as possible. He thought the universe would be better if Rey did not harness the Force, and this might speak to his fear OF her rather than his fear FOR her.
This theory depends on Luke upholding the "no love" clause of the Jedi oath, which puts a lot of faith in a young man who was tryna with his sister because she was the only girl he'd ever met. But this theory's got more evidence going for it than Luke's farm boy chastity. Let's look at Han and Leia's reactions to Rey.
Redditor noPENGSinALASKA describes the look Han gives Rey after she remarks that she didn't know this much "green" existed in the universe as one of "total guilt." Han would only feel guilty if he was somehow responsible for stranding her on Jakku.
Leia also has a special reaction to Rey. They embrace at their first meeting, speaking to a deeper connection, perhaps a mother-daughter bond?
So, why wouldn't Han and Leia tell Rey: "I am your father...and mother!"
Redditor Superego366 offers a possible explanation: Out of jealousy or spite, Kylo Ren wiped their memories and made them forget about their daughter.
The majority of theories about Rey's origins cast her as a descendent of Anakin, but this theory, outlined on 9gag, asserts that she is Anakin. As far as we know, extensive plastic surgery isn't a thing in the Star Wars universe ('cuz you know Palpatine would have been all about those face lifts) so we don't mean Rey is literally Anakin. This theory posits that Rey is the reincarnated Chosen One, and she must use this life to repent for past mistakes and to seek redemption.
This means that her sad childhood on Jakku is Anakin's purgatory. Although her rise to power parallels Anakin's (naturally good pilot, supremely force-sensitive, looks a lot like Shmi Skywalker), she will not succumb to the Dark Side and intends to defeat the Sith once and for all. Her powerful connection to Luke's lightsaber has nothing to do with being Luke's daughter; it's because the lightsaber originally belonged to Anakin.
Also, this theory explains why Kylo Ren can't communicate with his grandfather. Anakin isn't a Force ghost anymore; he's a young woman. And he's gonna teach Kylo a few things about the Force.