Yeah, I've managed to find a responsible friend to join me, and my mum should hopefully be okay with it (I'll ask her closer to the time) since she let my brother go on the train 5 hours away with a friend when he was 17, and i'll be 16 at the time so it should hopefully be alright!
Just sit down and talk to her. Tell her that you're a responsible person, tell her exactly where you'll be be, and tell her that you'll call her when you arrive, when you meet with your friends, when you're leaving to catch the train, and obviously when you get home. Keep communication open, because that's how you build trust.
You said you're almost 16. While it might seem a bit young to be travelling by yourself, you're also at the driver's license age and your mother can't keep you sheltered and within a 10 mile radius your entire life--otherwise, when you are able to get out, you won't be able to handle yourself in these situations.
Train rides aren't that bad. You'll literally sit for 4 hours. Not a lot can go wrong there. I see how she can be worried about meeting up with strangers, but that's up to you to portray your friends to your mother as smart, responsible people that you can be trusted with. As for getting home, you should be fine as long as your at your track 5 min before the departure time. If anything, you could catch the next train even though you would probably be waiting.
My parents were the same way, but in my situation, I wanted to travel long distances to concerts, which is arguably worse. But in the end, they decided that they had to trust me (especially because I would have found a way to get there one way or another). Sure, it's nerve racking your first time doing something like this, but it's so much fun! It's fun being out in the world by yourself! I ended up going to NYC and navigating subways by myself, and while I got on the wrong subway twice and missed my train home because the concert ran later than I expected (had to pull an all nighter at a friend's apartment that lived in the city and catch the first train back at 5 in the morning), it was a blast. Then again, I was 18 when I went through this but the first time I took the train by myself was when I was 16.