I do think the lack of uptake of STEM subjects by girls is a real problem, but I have no idea what the solutions are.
I can (and have) banged on about this endlessly, including on several MN threads. I think you're right about the education bit, as there are a lot of stereotyped images around what STEM careers involved, and they just don't cover the huge range of roles which are actually available.
Nearly every country (including the UK) has government-sponsored organisations to help promote STEM and forge links between educational institutions and industry. And women currently working in STEM careers are more likely to be aware of all the issues than their male colleagues. There's a limit to what anyone can fit in on top of your day job, though, and STEM promotion is usually extra to the job.
In any case, none of this helps the OP's daughter. I think it is worth asking why she's feeling intimidated, because there might be particular things going on which can be dealt with, such as bullying behaviour. I'd be wanting to know whether she was the only girl who was really interested in doing FM this year, or whether she was the only girl who felt strongly enough and was strong enough to sign up for it, when it's generally a negative environment for girls.
Some schools are better than others; some do have sexist teachers who still have a problem with girls doing STEM subjects. Some schools are far better at promoting STEM subjects to girls and supporting girls - it's not a coincidence that girls in single-sex schools have a greater take-up of subjects like physics and maths at A-level than girls in mixed schools, but even in mixed schools, teachers can help create an environment which is more encouraging to girls. I'm assuming that changing schools isn't really an option, except as a very last resort, but it may be worth asking the school how they deal with stereotype threat and implicit bias and so on. The very fact she is the only girl will be giving her the message that girls don't do this, and that in itself can cause her to doubt herself and affect her confidence.
I think support in terms of a mentor if you can find one is good. It's hard enough to find one at work, so I'm really not sure how you'd go about it in terms of a teenager at school; however, I am sure that there will be people out there willing to assist once you've found them. If you're in a university town, I'd try to make enquiries there, to see if they might have any support - even just an hour's talk could help reenthuse her. It might be worth asking your local Stemnet coordinator.
If it's just about her confidence, then maybe look at whether there's any assertiveness training available anywhere. Amy Cuddy's TED talk on body language could help, so she can learn to sit up straight, hold her head up, sit with her arms wider, so she's physically taking up more space. Learn to project her voice more. There's also loads written online about men talking over women in meetings in the workplace, and it's this sort of classroom where they learn to get away with this, and girls learn that it will happen. If you google subjects like "how to get your voice heard in meetings", there could be some useful tips there, too. Also, something I find useful to remember is that if I need to ask a question, I'm probably not the only one in the room who needs it; however, it will sometimes be difficult to believe that in a room full of boys who are full of bluster and not prepared to admit to not knowing. It will help if you've got the teacher onside.
I doubt any one of these things in themselves will fix it - but a combination of some of them will have a cumulative effect, plus if she finds her confidence growing, that in itself will help her even more.
Good luck to her - it can be hard work being the only woman in the room, but it's worth sticking at it.