I do those things too, but trust me, your mother sees it differently. Mine does as well. If she's with you, there shouldn't be that many safety concerns. Listen to Frost's advice about safety.
Here is a thought and project that you might want to consider, It'sMeStella. Do this first, as your homework before you talk to your mom. Go to YouTube and watch several of the videos of what went on at BronyCon in Seacaucus this summer. I have some idea of what went on because I was there myself, and can vouch for the fact that a mature teen girl who observes the precautions already mentioned on this thread should have no trouble or worries about attending. Actually, I was one of the cosplayers dressed as a Pony, as well as the fact that I saw many teen girls cosplaying as Ponies themselves, and having a lot of fun ! Pick out maybe four or five of the videos that show what it was like to be at BronyCon, and what was happening. The videos should show that the convention was just a gathering of bronys having a good time, and enjoying their mutual interest in My Little Pony by attending panels, dressing and cosplaying as Ponys, and behaving in an acceptable manner. Then when your mom has sufficient time to watch, sit down the two of you together, and watch the videos. Maybe stop the videos from time to time and discuss what is going on. After the viewing, also have time to discuss if going to the next BronyCon would be a good, fun thing for you to do. Also have a paper copy of the suggestions and precautions mentioned on how to be safe at BronyCon, and mention that you would abide by the recommendations. It may be that your mom would be impressed that you " did your homework " , and you already thought about the issues of you going. I hope that this suggestion helps in some way so that not only you can eventually go to BronyCon, but your mom can see that you are already being mature about presenting your thoughts and reviewing information with her.
I think that her main worry is that she doesn't know anything about bronycon. With the mall or movies, she must know the place quite well, so the worry isn't there. Bronycon is completely new to her, so don't take it like she doesn't trust you.
Wow, I guess I'm in a different boat than all of you. I can't stray away from my parents when I'm anywhere (not even the grocery store...) much less the mall or BronyCon. As much as I would love to go to bronycon, there is just no way I'll be able to go any time in the next 4 years. I'm just too self conscious of a person. My dad barely understood me why I watchde Dragonball (Which is rated TV14 which is exactly the right age group as I'm 14) I will just never have it in me to tell him I watch My little pony... I try to tell myself otherwise but I just can't do it. He'd think I was undergoing some kind of depression or something and try to use therapy to slap it out of me which would ultimately lead to me losing ponies and internet freedom. He and my mother are very cautious people, and I know it's a parent thing, but hey, they wouldn't understand why I watch the show, my dad wouldn't even try dbz or any of the games I play. I don't get any outside world support about being a brony, which is why I started spending so much time on this forum. Where theres nothing but bronies. But one day I'll be old enough to make my own decisions, and go where I want to go, then I can go to meetups and bronycons (assuming the series goes on that long, but with this much support, I'm sure it will). This has been my emotional post of the month everypony! ~Vegeta