Problems with my homophobic friend

I'm assuming that school started for you in September? That would mean you're almost done your first standard year (2 semesters) of university, if my assumption's correct. It's much too early to say that you 'have no one else'.

Are you living on campus? If you have a roommate, that's an easy way to make friends and acquaintances. Surely you've talked to others besides this girl, as well. It's likely that many others are feeling, or have felt, the way you do right now, in terms of loneliness. With that said, if you tried to engage a classmate in idle chatter, it's likely they'd allow you to.

An easy way to start a conversation is to talk about your courses. If you overhear someone having difficulty with something, and you think you can help, or you're having a similar problem, it's easy to just go over and talk with them a bit. That's actually how I made my first friend in university!

In terms of the girl's homophobia, the sad thing is that prejudiced people are everywhere, and picking fights likely won't solve anything (I'm not saying that you pick fights, of course). It's possible that she hasn't been exposed to positive exposure of LGBT+ things. Many people change their minds about things once they see things for themselves. In this instance, if she sees that LGBT+ people aren't savage blood-sucking vampires, maybe she'll change her mind, but that change would probably be gradual.

 

Since you mentioned a psychology class, I'm going to assume that you're studying some sort of psychology, if only as electives. And if not, just read on and you'll learn something new.

There's a phenomenon called "cognitive dissonance". Basically what it says is that when our actions don't match our beliefs, we experience a discomforting feeling called dissonance, and we have to change either our actions or beliefs so that they match up in order to alleviate the feeling. In this case, your friend both believes and acts upon (by telling you) that she doesn't like LGBT people.

Now, what happens if you were to come out to her? You'd question her opinion of LGBT people, since you're nice and her friend, but LGBT people are "bad" in some way. Despite her thinking that LGBT people are bad, she has acted opposite to that towards you (besides when you're arguing). This will cause her to experience dissonance, since her beliefs (LGBT = bad) do not match up with her actions (friend who is LGBT = good) anymore.

In alleviating dissonance, usually the person changes their beliefs because it's easier to do than changing your actions. Humans like to appear consistent by nature. She can't really "deny" to people that she's been nice to you, since well, she has been nice to you, and I'm sure there are plenty of tangible things to prove that. She can, however, change her beliefs, since who can really tell her what she's really thinking but her?

tl;dr That's a very long-winded way of saying that if you come out to her, there's a reasonable chance that she'd be accepting, unless she's really anti-LGBT.

Otherwise, I'd second what Luthan said. I'd wager you're uncomfortable with the idea of potentially losing your best friend (rightfully so). Could you start building up a support network of other people? Other friends/family you could come out to first?