MC will definitely believe his parents...right? wouldn't he believe the story about gracie after reading the letter from his parents? reaching out to gracie just after reading the letter only to find out she's counting her last days and MC got 3 daughters. that makes sense.
Also MC saw gracie a lot of times in his dreams so he would have recognized gracie and would believe the story.
From the main character's perspective, the name and face would be disconnected. Just a jumble of words and pictures.
Unless the letter contained a photo or a textual description of her looks there'd be no connection.
She used to visit him in hospital and write him letters. His parents would have known her from that.
He only knows about Gracie from his dreams - and only the face and the sound of her voice but not a name.
We're told he has no memory prior to the accident as a direct result.
Deep down, Gracie is still an important piece of him and through her he begins to come back to himself in his dreams.
This is why he reacts to seeing Liv in the way he does.
It's said that people in a coma can still hear you. I still think that the chance of him remembering Gracie's name is very low at best. Through her visits he could put a name on her voice but couldn't attach a face.
Anyway, I'm done now. I would like a "What if" scenario but I'll speak no more on this matter.