It's a tough situation to be in.
I remember being that age and feeling like I was old enough to take on pretty much anything. But actually, the internet can be a pretty messed up place sometimes, and I think that's probably what's worrying your mom (rather than it being an issue of not trusting you for example).
One thing you could do to allay her fear is by going through the "safety tips for parents" guide that Twitter has published here: https://support.twitter.com/articles/470968-safety-tips-for-parents
It will help your mom understand there are actually a bunch of things you can do to stay safe, report offensive profiles etc..
Another thing you can do to convince your mom is to highlight the benefits of social media: Being able to keep up to date with world events, find interesting and educational material, learn to communicate and market your ideas.
Social media is becoming an important tool for every type of business these days - so getting an early footing is very valuable.
On the other hand...
Many sites have age limits that prevent young people from being exposed to some of the material on their websites.
For example, Blurtit has a 13+ policy because some of the questions we receive are of a "mature" nature. We try to protect users who might be confused or upset by these kind of discussions.
It's an interesting question you've asked here though because, as the world becomes more switched on to "social media", it will only make sense that younger people will want to take part.
Finding an effective way of providing a safe experience for all is super important, and something we all need to work towards in a sense.
