SS is much more forgiving than say, the military or the passport office, and driver’s licenses come in two types now.
My name was misspelled on my birth certificate, and I had to live with the misspelling on my first passport, issued when I was 9. Having always used the spelling my parents intended, though, I got an SS card with that spelling, and a driver’s license with that spelling. So I initially entered the military with my preferred spelling, and used it all through training. When I got to my unit, however, some paper-pusher started changing it on all my paperwork, saying that whatever my birth certificate said was what I had to use. I pointed out that it would not match my high school and college transcripts, not for that matter my dogtags, and the military ID issued me at basic. She didn’t care.
So I went through the process of changing my birth certificate. This was New York, FWIW. It was free, but it required me to document that I had always used the other spelling, and to state in writing, and have notarized with my signature (free at my bank) that there was an error.
One thing I had that demonstrated that I had ALWAYS used the other spelling was my birth announcements. My parents had saved the extras. I had a xerox of my first grade report card, a cancelled check from the account I had when I had a paper route when I was 11, and so forth. I had to pay $10 for the new copy of my birth certificate, but it was worth it to shove it in that PFC’s face.
Anyway, I suppose you could claim that it was an error that caused her middle name to be left off her birth certificate.
Other than that, I think you will need a formal name change. Since 9/11, this is taken pretty seriously. I had a friend who changed her name after gender reassignment medication, and it cost $100, and a hearing before a judge. The judge wanted to see a doctor’s note, evidence that she had been using the name as long as she had been presenting as a woman, and anything else that would help. I went with her to tell the judge that yeah, everyone called her that.
I have a feeling that you will need a hearing before a judge if she is going to want to have this name on her passport or secure driver’s license. If she doesn’t care about those, and is happy to use it informally, and just have her first and last names on her passport and license, then she can just use it. Most people wish they didn’t have to have their middle names on their passport and secure driver’s license, but if you want to use them to board a plane, it’s required.