Rey's Origins

Leia and Han
This seems the most obvious of the theories out there, that Rey is the second child of Han and Leia. Here are the things that point to this rumor being true.
-Rey has a natural ability to pilot and fix star ships, especially the Millennium Falcon.
-Rey often says the same things as Han.
-The Millennium Falcon just happens to be owned by the guy that she was dropped off to by her family.
-When Rey tells Han her name Han offers to take her as his second mate on the Falcon.
-When Rey goes after Finn in Maz's Castle Maz asks Han, "who's the girl" and then cuts away, implying that Han knows who she is and that he has a connection to her.
-When Kylo Ren learns that a girl is responsible for BB-8's escape he freaks out even more than usual, implying that he knows who she is and has a connection to her already. He does the same thing when she escapes in the Starkiller Base.
-Kylo tries to persuade Rey to join him as his student in the Force, why would he offer to do that for a stranger?
-Rey and Leia seem to possibly know each other after destroying the Starkiller Base, they look at each other for a time and then embrace each other, despite this being the first time they've ever seen each other? Seems fishy.
-Rey is given the Millennium Falcon to go after Luke with Chewie, as if she has inherited the ship.
-Leia and Han have a boy and a girl in the EU books with the boy turning to the dark side and the girl being a jedi.

Things against:
-Han and Leia would be the worst parents in the world if they not only had Kylo become a psychopath but also abandon and completely forget about their second child on a remote planet and not recognize her when they see her again.
-Rey has a British accent, where did she get that? You would think that at least one of Rey's parents would have the accent as well, where else would it come from? Certainly not Jakku, it had to be acquired before she was abandoned.
-In a movie where both Han and Leia are featured a lot you think one of them would have said at least something to at least suggest they were her parents.
-Maz says that Rey must know that her parents aren't coming back to Jakku to get her. Maz also says that the lightsaber belonged to Anakin, then his son Luke, and now it is calling to her. This seems to imply that she is Luke's kid, not Leia's.
-In her vision there is no clues suggesting that Rey is Leia's kid.
-The lightsaber skips over Kylo and goes to Rey instead, as if she has a better claim to it than he (as Luke's nephew, not daughter) does.

Luke's child
This is the second main theory out there, that Luke had a child with some random woman, something happened and he abandoned her on Jakku. There are strong facts for and against this one.
Things for:
-Rey's adventure in TFA is nearly identical to ANH. They look similar. They both grow up on a desert planet that they are not originally from. They feel alone. They both search for a life with greater meaning. They meet an old war hero who becomes something of a mentor for the movie before they die. They are both introduced to the Force and given the exact same lightsaber.
-Luke has more motive of dropping off Rey on a deserted planet than Han and Leia because he abandoned "everything" when his apprentice turned to the dark side, that could very well include Rey, perhaps to protect her from Ren.
-Luke is shown in Rey's vision when she touches the lightsaber. They seem to have a bond through the lightsaber, just as Luke had with Vader.
-Rey tells BB-8 that her origins are classified. Luke is the number one wanted man in the galaxy, by both sides, and so having her true identity revealed to the galaxy would put her in extreme danger.
-Rey and Luke have an interesting stare off at the end of the film, some interpret it as it being daughter being reunited with father.
-Luke's theme song comes on when Rey reaches out with the Force during her lightsaber duel with Kylo Ren, in a movie that rarely used any scores from the original trilogy. This song could connect them.
-Leia's embrace with Rey almost seems like it could be aunt to niece after not seeing each other for a long time.

Against:
-Jedi are forbidden from having children. This one is a little weak as in the EU Luke has kids and the fact is that he can basically do whatever he wants because there is no one to enforce rules except for him. Despite this Luke could have taken steps to rebuild the Jedi Order to exactly how it was before the Empire with rules like this. Or Luke could have seen that it was because of this rule that Anakin fell to the dark side.
-Who is the mother? There are no indications that Luke ever settled down with anyone. There is no one in the original trilogy and no mentions of a girl in TFA. It seems like Luke was only motivated to rebuild the Order, not to settle down. And if he did marry, where is the mother now?
-Again where did Rey get her accent? It is possible that Luke's wife had an accent or raised her in a place where she developed it because of her surroundings, but it seems a little out there.
-Rey seems genuine when she says to Finn, "Luke Skywalker, I thought he was just a myth?" Why wouldn't she know that Luke is real if he was her father?
-Rey appears to know nothing of the Force when we meet her, if she was Luke's daughter you'd think she'd at least know a little about it.


Obi Wan Kenobi's granddaughter
I am leaning the most to this theory recently. The theory is that Obi Wan, after settling on Tattoine to be near Luke, had a child with some chick. He was there for like 20 years and you'd think old Ben would do something besides scare Sand People, maybe he settled down with a girl for a time. They had a child and then that child eventually had Rey.
Things for:
-This could explain Rey's accent as Obi Wan is one of the very few characters that are good to have a British accent.
-Obi Wan was actually in possession of Anakin's lightsaber for more time than Anakin and Luke combined while he kept it on Tattoine. Maybe the connection is not to Luke but to Obi Wan instead.
-In Rey's vision we hear the voices of the two Obi Wan's as they say: "Rey, these are your first steps."
In the vision we also here Luke yell "No" like in Empire and Yoda talk about the Force being all around us. Both Luke and Yoda's talking is general in nature and only repeats of what they said in previous movies, but Obi Wan is the only one that really directly speaks to her, telling her that she is following in his footsteps.
-Despite knowing next to nothing of the Force she busts out a jedi mind trick to trick a Stormtrooper, almost exactly how Obi Wan does it in Episode IV. Besides Obi Wan we only see Qui Gon Jin try to use the jedi mind trick, and he fails at it! Each jedi has a particular skill that they are known for or best at and it seems like Obi Wan will be remembered most for his mind tricks.
-Rey sneaks around the Starkiller Base just like Obi Wan does in Episode IV aboard the Death Star.
-It could be that Rey's parents knew little of the Force and were terrified when their daughter had great Force powers, and decided to drop her off on Jakku in fear of her.
-Daisy Ridley, the actress who plays Rey states that Rey's solitary nature is a big clue as to who she really is. Some interpret this meaning that she flies "Solo" but it can also be interpreted as being just like Obi Wan was for 20 years living in isolation on Tattoine.

The things going against this theory are many and would require a lot of backstory telling to explain it all.