What Can I Do To Convince My Mom Into Letting Me Have Text Messaging On My Cell Phone?

You don't, unless you start acting like you can handle the privilege of what she has already given you. She may not want the added expense on the phone bill for texting and something tells me you are not being totally truthful with her about the texting of messages you are wanting to text. If you want the privilege then you should prove thet you can be trusted with other things that she gives you to do and things she has given you. I do not know your age, but if you are old enough to want a  phone to text on then you are old enough to help pay the bills for it. Like the old saying "if your friends jump off a bridge would you do it to"? Or if you want it earn it.then you will take care of it because you paid for it unlike your friends whose parents are probably paying the bill for them.

Show her the text messages

First off, remember that parents don't see things our way, so an argument of "all my friends are doing it" and "there was this cute guy who wanted to text me" doesn't make em' bat an eyelash. Think about what she wants to hear. Is she worried about money? Ask if you can do work around the house, and if you shirk your chores, she can take the privilege away. It's about respectful negotiation. Don't whine, act serious and sincere, not desperate! I know not all parents listen, but I've found this one to be better in most cases...

Best thing to do ever.
Find a friends mom that has verizon.
Give her the money for the phone and the add a line fee.
Then pay her for texting a month.
This way when your mom gets mad at you, like mine just did today, she wont turn your phone off.