In Edge of Tomorrow, Rita Vrataski dramatically tells Cage her true middle name before one of her deaths. When I was watching, I assumed that Cage would use this knowledge to gain her trust in a crucial moment later on. But no such moment occurs, and the name is never mentioned again in the movie. So...what gives?! Was there a scene that was taken out that used it? Did it come from an earlier script that was later edited, but not edited enough?
As some of the other answers have suggested, it might really be an open-ended part of the story where the explanation was dropped at some point to save on screen time or because the story was changed somehow. That doesn't necessarily mean that we can't find meaning in it within the final product that we see on the screen.
Speculation: it was a tender moment that shows that despite Rita's stony approach to Cage, and despite the dire situation they were in, they were able to connect in an intimate way. The virtue of the act is that Rita gave something to Cage that he she had been arbitrarily withholding from him, opening up to him. It showed that she cared, not that she was planting some secret password for her to automatically let him into her life.
It was character development, not plot development.
The middle name is not the point. The point is that, for whatever reason, Rita does not like others to know her middle name. We can speculate what that reason is - I would guess that "Rose" clashes with the hardened image she likes to project - but, in the end, she does not want some random guy to know that name.
Her telling Cage her middle name has nothing to do with the name's innate importance. It's there to give the viewer the exact moment when she decides that he is not just some random guy - shortly after the scene with the coffee, when Rita realizes that, to him, she is not some random girl.
When he tries to protect her any way he can - even by lying, possibly at his own expense - she feels that he has the right to know this otherwise small thing he's been trying to coax out of her. And, since (in her mind) she'd only known him since morning, we can also gather that very few people treat her that way, and that she is actually desperate for this kind of genuine connection.
Which, of course, means that her public image - Full Metal Bitch - is really just a psychological wall she built, possibly to distance herself from the adoring fans who worship her for Verdun, which she sees as a profound failure.
The only screenplay I can find that is based on the shooting script is transcribed, and terribly so. It's here.
But I did find the first draft titled All You Need is Kill. It is a GREAT read and the storyline is very, very different. I'm not all the way through it yet, but the first draft has no "middle name" scene, and when you read the scene leading up to the controversial kiss (pg. 96 - 97) it makes a lot more sense.
Basically, it was an impulsive move by Cage after some intense, adrenaline fueled battle that was answered by Rita with equal gusto. Kind of a "Oh my god, we're still alive!" moment. When you consider the huge difference between the scripts added to the constant resets, it throws this whole middle name business on its ear.
My best stab at "Who cares about Rita's middle name" is: Exactly! It appears to be some producer's lame patch job after changing so much of what I think would have been a far better story.
This is the event that causes Cage to alter his approach, as he comes to terms with the futility and emotional toll of the current approach.
What came before that
Just before this scene, when he and Rita are talking, the topic of a man named "Hendricks" comes up. Rita eventually divulges that she saw the man die some 300 times, and remembers every moment of it. It is implied that she was in a romantic relationship with Hendricks, and in any case she was scarred and traumatized by watching him die so often.
What came before that
Cage has been trying to get Rita closer and closer to the Omega many, many times. We're never told how many. We see clips from a few dozen at most, but we know for a fact we haven't seen every iteration (we learn that he's gone through the helicopter part many times even though this is the first time we've seen it), so it may have been hundreds, maybe even thousands. In many of them that we do see, the clip ends with him watching Rita die, and then he dies as he grieves or just otherwise gives up and waits for a reset.
Sounds an awful lot like the Hendricks and Rita thing, doesn't it?
What came after "Rose"
Cage again somberly gives up, seemingly in grief, and gets killed. On the next iteration (that we see) he abandons the idea of involving Rita and tries to get to the Omega on his own.
Altogether we are led to conclude that Rita is Cage's Hendricks. He's attached to her, doesn't want her to die, is tired of watching her die over and over again, and is powerless to help her (or anyone else) by any means other than abandoning her and working on his own. The scene where she says her middle name is "Rose" just before dying isn't there to give Cage more information to use, but to trigger a change in his resolve, tactics, and dependency (he was, up to this point, acting entirely on the assumption that Rita was the one that would bring things to a final resolution, not him). He finally takes ownership of his situation, and makes the tough calls that he needs to make.
Personally, one of the first things I was thinking when he first started trying to get both of them to the Omega was that she was dead weight: he had to spend significant amounts of time just trying to prepare her for the exact sequence of events they would have to deal with, and trying to find alternative paths that might work better. I was thinking he needed to just ditch her and focus on getting himself through. But he was dependent on her. She had the resolve, certainty, and will to kill the Omega while he was still trying to adjust and understand what was happening. It took a long time before he came to the conclusion that there were casualties he couldn't prevent (we see him stop trying to save Mr. Balls Out, for example), and that he had to triage things: some deaths couldn't be stopped if he was to achieve the greater good, and he had to do this on his own.
Up until the next plot twist, anyway.
Some critics have criticized the kiss at the end of the movie as coming out of nowhere and not being true to Rita's unemotional, business-like demeanor. Perhaps the middle name reveal was a preemptive move on the part of the writers to show that underneath her cold exterior, Rita had feelings for Cage (demonstrated by her sharing this personal secret with him before she died).
Rita fakes her middle name every time. He relived her death and then tested what she told him during the car ride the next time. He said that she would tell him more personal information, such as her middle name, up ahead. She knew that he would wake up in yesterday and recall what she had told him, so she gives a new fake middle name. It's a joke of sorts.