It’s not asking to called by a boy’s name that concerns me most about this young girl. It’s your report that she is angry, isolated, friendless, and miserable.
Somehow I think the name change is a red herring. There’s something far more serious going on. Is there any possibility that she was hurt by a man and doesn’t feel she can tell anyone? Sometimes wanting to be male is a way a girl is saying that she wants the perceived power of a man so she can’t be hurt by other men. Sometimes it’s a way to try not to be female and vulnerable. Sometimes making demands on others to call her by a boy’s name is a girl’s way of saying “please notice that something is wrong and help me.”
I’d be more inclined to see this as a “phase” if she seemed happy with herself and her life. Since that is not the case, I think it would be wise to address her anger and unhappiness directly. Sometimes a kid will respond to simple, compassionate and kind questions. Please put the name issue aside for a bit and let her know that you are worried about her, that you are on her side, and that you will stand by her no matter what. You can tell her that you’ve found out that sometimes kids who make a request like hers are really signaling that something is terribly wrong and if that is the case with her, you want to be there for her. Let her know you don’t care about a name, but you very much do care about her. If you are unsure about how to go about this, please go see a counselor who specializes in adolescents to get some coaching. It’s very, very important to treat an issue as serious as this may be with great tact and care. You love this girl. With a little support, I’m sure you can help her.
I wish you well.