How can I prove to my mom that I'll be responsible on Facebook, and that it's a safe website?

[Laura]

When you make a Facebook, you can change around your privacy settings so that
1) you can set it to where only friends may see you profile and pictures,
2) You can change your settings to where your name does not even show up if there happened to be anyone searching for someone with the same name as you (which is extremely likely)
3) Facebook is not like a chat site where anyone can chat to anyone else openly. You may only chat openly in disscusions from fan pages or group pages. And even then you are actually only POSTING something and not actually chatting, as seeing replys to your post requires refreshing the page. You can also only chat to those who are your friends and if someone random happens to request you as a friend, not only can you ignore them, but you can also mark them as unknown.

All in all, Facebook is pretty safe.
Hope this helps