I see several ways it could play off.
What is extremely unlikely
1)A character from the new Extended Universe books, comics or videogames. Battlefront II is heavily hinting about that, but IMO this is misleading (the lukewarm reception of Battlefront II was previsible, and bland characters from a MP shooter that use them sparingly in the SP campaign are unlikely candidates ). Both in the Old and the New EU, entire book series were written about characters with one line or two in the movies, the reverse not being true.
What is possible but cheap
2) A character from the movies or the TV shows that could have fathered/mothered a child, more or less naturally, but that is not a Skywalker. That means a Human. From the mommy side, we are almost down to Mon Mothma (who is kinda the only Human female in the OT except Leia and ''girl that operate the ion cannon on Hoth'') or Sabine. For the daddy side, there is a very insidious Sith that might have enjoyed some simple pleasures except ''cackling'' and ''being super-duper evil', but he croaked 11 years before Rey birth. While siring someone that would wreck vengeance on Vader progeny is typically him, I presume that someone at Disney would have pointed out that ''finding in an Emperor secret base a vial filled with a white fluid'' had strong potential for mockery...
3) A Skywalker, which would be extremely cheap. But I mean, Luke was the Galaxy most eligible bachelor, are we supposed to believe that he remained a virgin until he was in his late thirties (thank you for too much information old EU, who came up with this pearl before ''Jedi are celibate'')
4) Another known character that is not a Skywalker. We have quite a collection of one scene wonders that require Wookiepedia searches (Jan Dodonna ! That sneering Intelligence officer from Rogue One ! The second patron to the left of the band in the cantina ! Lobot ! ) and Lando/Han (both men of the galaxy to say the least). But Lando is not a very probable candidate for even the most open-minded color blind casting. Han ? That would sounds jncredibly cheap, but it could explain Kylo Ren daddy issues and could be neatly tied in Solo the movie for people aghast at inventing an off-screen girlfriend or sister of Han Solo.
What is very possible in universe, but unlikely from a movie POV.
5)Rey is like Mara Jade, AKA her origins are simply unknown. She is not the grand daughter of a master Jedi that appeared for six seconds in ROTS before being mowed down, her powers are from no one. She is no one, and that's what enrage Mister Mediocre Jedi and Mediocrer Sith, whole like Draco Malfoy considers that he is the heir of a Force lineage.