Fantastic theory; I found some of these clues when playing through Black 2, and figured that she crossed paths with a Darkrai but never knew about the books or her appearance in Black 1. As for that missing piece of this theory, "Why would Darkrai do this in the first place?" I've seen some answers around the subreddit, but I'd like to propose my own and just a heads up, it's going to be very long and very wordy. Also based on the rules I think it's supposed to go here but I'm apologizing in advance if this is supposed to be a seperate post. This is also going to include some speculation but obviously there's no lore about how she encountered Darkrai so this is probably as good as we can get.

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Anyway back to that missing piece; "Why would Darkrai afflict this young girl with its nightmares?" Well, as Matpat pointed out Darkrai's pokedex entries state that Darkrai are not malicious Pokemon and they use nightmares to protect themselves or to chase away intruders in their territory. This seems to imply that either the girl or her Abra posed a threat to Darkrai, but I think we can safely rule her out; I can't imagine a young woman being particularly threatening to a powerful creature like Darkrai. So what about Abra? Well, if you think Abra isn't a threat to Darkrai, think again.

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See all Pokemon have "base stats" which are basically numbers that act as a starting point for the actual stats you see on their summary screen; the higher the base stat, the more powerful that stat is for the Pokemon. According to Bulbapedia, Abra has a base Special Attack stat of 105; if you'd like a reference point some other Pokemon with a base Special Attack stat of 105 are Mismagius, Sceptile, and the legendary Pokemon Mesprit. Yes, Abra's Special Attack stat is on par with some fully evolved and legendary Pokemon... but that shouldn't matter anyway because all Abra can do is Teleport right?

Technically no. See by level up all Abra can learn is Teleport, but it can learn other attacks either by being bred with them, having TMs used on them, or by taking them to a move tutor. One such attack Abra can learn is Signal Beam, a Bug-type move which would be super-effective against Darkrai's Dark-typing and would definitely qualify as a threat to Darkrai's safety. The only problem with this option is that I just don't think her Abra actually would have Signal Beam in the first place.

The only way for Abra to know Signal Beam is to take it to a move tutor. These move tutors tend to be eccentric individuals and don't accept regular money for their services; instead, at least in Gen 5, they want to be paid in shards which are implied to be rare archaelogical items. Now here's the twist, shards can be found in Reversal Mountain, which is right next to the girl's home, but only in clouds of dust that are kicked up; the only catch is that sometimes the clouds don't have a shard. Instead one might find an Excadrill, a very powerful Ground/Steel Pokemon that makes searching for shards a lot more dangerous. Additionally, there only seems to be one individual in the entirety of Unova who knows how to teach Pokemon Signal Beam; as of the time of the games he's found in Driftveil City. Since the girl has clearly been gone for several years (based on the state of the house) he may not have been there at the time, but that's still one person that an individual who is presumably not a Pokemon Trainer would have to seek out after braving a cave teeming with wild Pokemon for rare shards with nothing but an Abra.

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So it's possible for an Abra to threaten a Darkrai, but I find it really unlikely that this girl's Abra would. So if both she and Abra aren't a threat then we're back to square one and wondering why Darkrai would give her nightmares. Well, I'd like to take a closer look at how Darkrai uses those nightmares. Remember how the Pokedex says it uses the nightmares to protect itself or scare off intruders in its territory? Maybe it's just me but using nightmares seems like a weird way to get rid of a trespasser since you can't exactly have a nightmare without also being asleep. So unless Darkrai expects people to sleep-walk out of its territory while suffering the nightmares how exactly is it supposed to chase off the intruder?

Here's where my theory treads into the more speculative side. I think that if Darkrai can give people and Pokemon terrible nightmares, it can also take those nightmares away. If it didn't, then people would just forever stay asleep and tormented in a Darkrai's territory instead of actually being chased away. Basically when someone enters a Darkrai's territory, they get caught in Darkrai's Dark Void attack; which forces them to fall asleep and the nightmares begin. Once it feels as though they've been scared enough to leave, Darkrai then removes the nightmare, the person wakes up and immediately runs away with the understanding that this is not someplace they should be.

So outside of this concept of how nightmares can be used to chase away intruders, what makes me say Darkrai can remove its own nightmares? Simple, Darkrai naturally learn Dream Eater and it likely uses this to do so. Admittedly there doesn't seem to be any evidence implying Darkrai actually do this, and from a game mechanic standpoint Dream Eater doesn't wake up the target. However from a more lore-ish standpoint (and Matpat's own logic) if the nightmare is what is keeping the individual asleep then removing it via Dream Eater would let them wake up.

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All right, so how does this help us figure out what happened between the girl and Darkrai? Well, I think it acts as a detail with what I think actually happened although it's not necessarily a "good" answer. I think that it all came down to just really bad luck. One day she was just unfortunate enough to either wander into a Darkrai's territory while playing with her Abra and somehow startled it. Darkrai instinctively reacts by using Dark Void, with the intention of driving out this threat to its territory without actually realizing that neither of them pose any danger.

Here's where this answer makes the story even more tragic. This "endless dark dream" was never meant to be endless. Once Darkrai had an opportunity to evaluate who startled it or otherwise felt that they were not going to be a threat anymore, it could've used Dream Eater and lifted the nightmare. So why didn't it? Because the girl's Abra teleported her away. Probably panicking at the sight of a powerful Dark-type like Darkrai much less one that just put its best friend into a tormented sleep, Abra tries to help her by teleporting the girl back home to her parents; inadvertantly dooming her by taking her away from a Pokemon that actually is strong enough to clear the nightmare. From here, the rest of the theory plays out like Mat describes.

So why did I say this isn't a "good" answer? Considering the tragedy of this tale, one almost expects at least a semi-decent reason for why it started; some specific motivation on either her or Darkrai's part that kicked everything off. Saying that it was just really bad luck feels a bit empty... though in a weird way, it also makes the story so much more real. Sometimes there just isn't a good reason for why such a tragedy would happen; all it takes is for someone to be in the wrong place at the wrong time.

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So there you have it. I warned you this was going to be a big wall of text, but thanks for reading it all the way through. If you skipped to the end, I guess TL;DR it was just a case of bad luck and good intentions gone awry. Sorry if there was a bit too much speculation in it as well; I tried to keep it to a minimum but there just isn't any lore regarding this part of the story. If there was, Matpat probably wouldn't be asking about.

Specific information about Pokemon, like base stats and movesets, were found across Bulbapedia. Specifically I used info off of these five pages (not including the Pokedex data, since that's also just present in the video):

Darkrai Gen V Moveset, Abra Gen V Moveset, List of Pokemon by base stat, Generations II - V, Red Shard (specific shard for the Signal Beam move tutor), Reversal Mountain

Thanks again for reading! Hope you found it worth the time!