Its a responsibility. Show her that you're responsible and have your priorities straight. Are you doing good in school? If you can do better, tell her that you will better your studies. Help around the house, and do work for her. Express how important it is to you and that you are only going to be this young once, and that you would like to do this while you still are young. If she says no, don't push the issue. It will lessen the chances of her saying yes in the future. Let it blow over for a while and bring it to her attention again here in a few months.
why would you want to pierce your belly button? Anyways, if you want to pierce it that bad then just try to suck up to her. like doing chores around you house/aparatment (wherever you live) or just pierce it without her knowing. good luck:P
You could try explaining in a mature way why getting your belly pierced is so important to you. And if it helps, explain to her a plan you can stick to afterwards to keep your piercing clean and infection free. Of course, this isn't a foolproof plan. Some people are extremely stubborn, so your mom might still say no. I hope it works out, though.