Parents

The obvious conclusion I think most people have drawn, is that her father is Luke, and the mother is someone that may or may not be important. The interesting twist is, could she be Luke's daughter, and Luke doesn't know about her existence? If Luke does know about her, do Han and Leia know?

Second feasible option, she's Han and Leia's daughter. I think this is less likely, because you would think that they would have made a bigger deal about her than they did. Especially with their concern for Ben.

Not quite sure this is unlikely. The first version of the script, written after Lucas's concepts, had a trio of cousins. Abrams/Kasdan started anew, going in another direction. Then recently we have Lucas explain that people often forget but SW at its core was a soap revolving around family issues, but the new team were not interested in his ideas and decided to go elsewhere. From Lucas's comments, it seems likely they've veered away from having a family saga at the core, and that disappointed him.

If Rey is Luke's daughter or niece in a trilogy that already has his nephew, I'm not sure how to reconcile this with Abrams's comments on the early scripts by Arndt, or Lucas's.

The reveal that Rey is a Skywalker or Solo is also what everyone is waiting for, so it's not necessarily the sort of things you want the audience to anticipate for two years, with the expected big "duh" effect at the arrival line.

Most of all, there are the circumstances. Leia and Luke had to be hidden from Vader and Palpatine for very good reasons. Rey was anywhere between 5 and 8 when she was abandoned on Jakku (but probably younger, or she'd have clearer memories of those years). Kylo would have known about his sister or cousin, and without Luke to train her as Ben and Yoda did with Luke, she isn't such a big threat.. In this context, why would it be the best option to abandon her to her fate, to try to survive as a child on her own on a most inhospitable planet instead of keeping her under the protection of Leia's Resistance or with Luke himself? Doing this in that context is just... horrible, whether Luke or Leia did it.

There are really three options with Rey: either the person who left her on Jakku thought he'd (or she'd) come back very soon for her but was never able to (the person probably died) or the person who did it knowingly abandon her there, with no intent of ever returning. The third option is one of "last resort".

In the last case, Rey is most likely a "padawan', a survivor of Kylo's attack on Luke's academy, and the person who left her there was possibly already being hunted down by the the Knights of Ren. Stopping to dump her on Jakku was a last resort, her only chance of survival before Kylo caught up with her guardian's ship. If by a miracle the guardian escaped, he/she could return soon for Rey... which didn't happen.

If Rey was abandoned, we have to look for people who aren't necessarily evil, but at least rather selfish: after what happened at the academy, they didn't want a child with force sensitivity and they dumped her. Not the most likely scenario. Or maybe they hated Jedi, but couldn't bring themselves to kill a young child. Not too likely either.

The "Jedi trainee" scenario is probably the most likely of all. It turns the story of Kylo into a personal tragedy for Rey since his actions against Luke also pretty much destroyed her life, and creates a connection to Luke without a blood tie. It might spark a conflict if Luke decides at first that Rey shouldn't be trained, especially not if it's to take revenge on Kylo. If no one but Kylo knows what happened to Rey's saviour/guardian, her inner conflict also remains unresolved, until/unless Kylo makes the big revelation to her that he killed her parent/guardian and thought her dead.

That scenario is certainly possible in the light of the novelization. There's this scene during the duel when Kylo seems to realize who Rey might be, but Rey doesn't recognize him (as she might a close relative). The same novelization makes some of your scenarios extremely unlikely, as we have Leia's thoughts on Kylo's history, what she kept from Han about him, what lead to Luke vanishing and Han and her to split up. Nowhere in there is a hint of a second family tragedy involving a little girl. It could be a cheat, but it would be a really big one. It at least rule out any possibility that Rey is a solo child thought to have died or disappeared from Kylo's attack on the Academy. That tragedy would be as great to Leia and Han as losing Ben, and it's not possible Leia would have made complete abstraction of it while thinking about the events surrounding Ben's turn to the dark side.

A big problem with her being a Solo or Skywalker is also her age. She was born many years (maybe 5 or even close to 10) before Ben turned to the Dark Side and the coming in the open of the First Order. It's very hard to see why her existence would have been kept a secret at the time while Ben's wasn't. That leaves only the possibility that even Luke doesn't know she's his daughter, but why wouldn't he? I don't find it so likely that Luke had an illegitimate daughter he never knew about, and even less that the mother or guardian would after he vanished think it's suddenly a good idea to dump the child alone on a junkyard planet...

Not only this would be an horrible thing for Luke to do, but why would he think a young child would be safer there on her own han with him, who intended to disappear?

This sounds very unlikely, unless the mother escaped from the academy with the Knights of Ren on her heels and she had no choice but to dump the child somewhere before they caught up with her... but why in such a scenario would Leia not know about her niece being missing or dead and not even think of this when she outlined the reasons why Luke vanished (in the novel)?

This is the option that makes the least sense. Why would Leia abandon her daughter on her own on Jakku, when she or Han have this vast network of contacts to make her disappear "safely", not to mention she'd probably be safest with either of them, especially Leia!

That doesn't work if she's a Skywalker or Solo, for some of the reasons above, and for the fact this, not Kylo's turn to the dark side would be the big tragedy for Luke. There's also the problem that Luke or Leia could probably sense if their daughter had died or not.

The most likely option, but this works much better if Rey isn't a Solo or Skywalker, and thus just another victim of Ben, not a personal tragedy in Luke's or Leia's life that it makes little sense not to ever mention even in the movie, and it's even more difficult to accept in the novelization where we get Leia's thoughts and Han's on the back story.... that's something HUGE to leave out, a massive cheat.

For these reasons I think the most likely scenario is that only Ben, grand-child of Vader, aside from Luke and Leia, will be from the Skywalker clan in the trilogy. Rey is something new, and for all the speculation I wouldn't rule out the possibility her back story is something we don't have the clues to puzzle out at the moment, that they will completely surprise us with something from the left field.

In the days of Lucas (that includes even the old EU), I would have said this is impossible, since SW is a family saga, but even Lucas now hints they've turned away from this, so...