Should she start to seriously considering it, that is going to be my argument. That, and probably about time I tell them about this and all the other sites I'm on. Considering that they are the main source of not only entertainment but social contact and a creative outlet (fanfiction, roleplay, etc.), I would hope to convince her otherwise. There is also my dad who I can turn to, and should I fail to convince her the first time, I could hopefully convince him and he would convince her.
But again, I currently don't think she would because she does use the internet. Besides, with my sister and I still in school, we need it for research unless she wants to drag us to the library everday to do it, especially since the 'just use books' argument wouldn't work well when you consider that the library is pretty small. Add in that we already have problems with scheduling due to my sister's tennis and my volunteer work, a research project would be a great way to convince her we need to have the internet.
But again, she probably won't get rid of it.