Who is Rey’s family?

We don't know.

There is nothing in the movie, nor the wiki's or official Star Wars page to suggest exactly who her family is yet.

There are many speculations that she could be Luke's daughter, Kylo Ren's twin sister, and as the other answer suggests (Which I really like actually, props for this one @John Smith optional) created much like Anakin by the force.

Summary

We don't know, but the movie hints strongly that she is Luke's daughter.

Explanation

The movie does not give sure evidence about her family, nor do any of the other current official Star Wars media like video games and comics.

However, the movie gives very certain hints that she belongs to Luke. The best hint and piece of dialogue is just after the vision Rey has in Maz Kanata's castle:

Rey - "What was that? I shouldn't have gone in there"

Maz - "That lightsaber was Luke's, and his father's before him and now it calls to you!"

Rey - "I have to get back to Jakku"

Maz - "Han told me" (they hold hands) "Dear child, I see your eyes - you already know the truth. Whomever you are waiting for on Jakku, they're never coming back" (Rey cries) "but, there's someone who still could"

Rey - "Luke"

Maz - "The belonging you seek is not behind you, it is ahead. I am no Jedi, but I know the force. It moves through and surrounds every living thing. Close your eyes, feel it. The light. It's always been there. It will guide you. The sabre - take it."

Rey - "I'm never touching that thing again, I don't want any part of this"

Some theories

1. She is Luke's daughter. I believe this because of the spoken sequence quoted above, plus she grew up on a desert just like Luke and Anakin wearing the same clothes as Luke and Anakin, she was poor like Luke and Anakin, she is a natural fly-er and mechanic just like Luke and Anakin, and the way Luke looks at her as compared to the way Han looks at her. Han seems to feel guilt that she grew up hungry, alone, and in a desert, multiple times (watch the scene when they arrive on Maz Kanata's planet and she says "I didn't know there was so much green in the whole galaxy"... Han's face says lots of guilt), but Han clearly sees Kylo Ren as his son and the way he sees Rey is very different. The way Luke looks at her at the end, however...
2. She is Kylo Ren/Ben Solo's sibling. The no-longer-official (no-longer-canon) Extended Universe books told a story of Han and Leia's twin children. The son turns to the dark and the daughter defeats him. JJ Abrams and company obviously knew about this and are using it to make people assume one way or another.
3. She has no family or she belongs to someone else. This is probably not correct. George Lucas's question to JJ Abrams in interviews is "What happened to Darth Vader's grandchildren?". The Star Wars films that have "Episode" in the title will all deal with the Skywalker family.

..is anyone yet to consider that

She may not have a 'family'?

Remember the immaculate conception of Anakin Skywalker's birth? he had no father, he was a being born purely of the force, and of the love of his mother.

Whilst it stands to reason that Rey must have a mother in order to be born, the hunt for who her father is may be a deliberate misdirection. In an age where the First Order are leading a crusade to find and destroy any remaining Jedi and Jedi Artefacts, it would also explain why whoever was her 'family' abandoned her on a distant planet, to hide her from 'herself', as it were.

Personal theory, purely conjecture and just to throw the cat among the pigeons:

Kylo Ren, being the grandson of Anakin Skywalker, has been raised with the notion of 'Destiny' surrounding his lineage. As Anakin didn't bring balance to the force, and (possibly) neither did Luke, he comes to believe that he is the destined 'Chosen One'.

He is the Star Pupil of Luke's Jedi Academy as a result of this: until, whilst Kylo is still a prepubescent, another of Luke's students (which, sure, could be revealed to be Jyn Erso for the sake of neatness) produces an immaculate baby alá Shmi Skywalker: this is kept relatively hush hush, for the childs protection

Over the next few years the attention shifts towards this new child: until eventually, as a moody older teenager, Kylo Ren basically rampages, and begins to massacre the younglings of the Jedi Academy indiscriminately to eradicate the 'chosen one', in a reference of Anakin's Jedi Temple attack on Coruscant.

Rey is hidden away on another planet, and as in the Obi-Wan tradition appointed a guardian to look over her: possibly Lor San Tekka, who is possibly portrayed in his younger years by Mads Mikkelson in Rogue One.

Considering the 8 Years between actors, and allowing for generosity, we see Rey is about 5-6 when she is 'abandoned' on Jakku. If Kylo Ren is circa 10-12 years older, this would put him about 16-18 when he turns to the dark side.

Skywalker goes into hiding, one of the only remaining people who knows whether Kylo Ren killed the correct child or not, and largely ashamed that he repeated Obi-Wan's mistake of not realising his pupil was being corrupted by the dark side.

Some of the minor details (like Lor San Tekka and Jyn Erso) are long shots and are an attempt to enact a cyclic narrative, but apart from those elements, anyone have anything concrete that completely eliminates this theory?

So, I guess that there are two ways to approach this:

In the movie, we see her using the force to reject Kylo Ren’s attempt to extort information about BB-8 and the map that he brings to find Luke. She could be the daughter of Luke Skywalker, or she could be the sister of Kylo Ren (son of Han Solo).

In the expanded universe Leia and Han have four children, but that’s no longer canon. Anyway, I think that she’s got Skywalker blood, because she uses the Force (and the lightsaber) and only a Jedi (or a familiar of that) can use it.

Although I find all the previous answers likely, I want to add another point of view.

They still have to decide it

A thing that I really appreciated of this movie is how characters are presented and how their past help us to understand who they really are. It may sound trivial, but I like the idea of a growing villain instead of the classic bad guy that comes out of nothing and dies barely speaking (hello Darth Maul). I also find the relationships between the various characters deeper than in previous movies.

It is my belief that they wanted to give Rey a deeper personality and in the same time presenting similarities between her and Anakin.

The choice of hiding her past leaves many options:

• it could be statically used in future movies to better define her personality, and since it didn't effect much the first movie of the third trilogy they are free to surprise;
• they could ignore it, like they did with Anakin, and keep going without leaving plot holes;
• it could be the chance to introduce other Jedi or other important characters that will be later presented (I just hope we are done with the "I'm your father" thing);
• they may just have took this decision to let fans speculate and dream about the possibilities while they wait for the next movie.

I feel like Disney doesn't have a specific idea of how things will go on and since they are not doing a prequel they are free from many restrictions. That could explain why they left many thing open such as Kylo's betrayal and Rey's past.

My guess is because she has the force she could be han and Leia daughter or she could be Luke's daughter. When Luke and Rey meet up it seems like they have a real connection with each other