How to convince my mom to let me get my belly button pierced again?

So, almost exactly a year ago, I got my belly button pierced. I thought it was really cute, and I took really good care of it. After about a month of having it, I got a bad rope burn on the inside of my elbows, which I think infected my blood or something, because my belly button started to get infected. I went to the piercer and he took it out and had to squeeze out nasty stuff, (which was gross and hurt like a b****, even worse than the actual piercing. I decided not to get it put back in because it hurt so bad. Now it's a year later, and I really want to get it redone. My mom doesn't want me to, however. She said she needs "time to think," so how much time should I give her? How can I convince her to let me? I even told her I would pay for it and take even more care of it this time. Also, if you have any thoughts on it, please feel free to comment/answer. Thanks so much!