The more you argue your point, the more she is going to *want* the tattoos. You are only reinforcing her idea that tattoos are rebellious and cool.
You don't have to encourage her to get a tattoo or anything, but I definitely would stop trying to convince her against them. She already knows tattoos are not your cup of tea. If anything, I would say to her, "Yes, you are going to be an adult soon and will be able to make your own choices. My hope is you will do your very best to make the right choices as you will be the one to live with the consequences of the choices you make". Then leave it in her hands. After all, the best way for us to learn is by experience.
When I was a teenager, most of my friends were very much into body modification and tattoos. Several of those friends either tattooed themselves or lied about their age in order to get very large tattoos. One of my friends even burned off a very ugly tattoo she gave herself with a curling iron! I would just be glad she has not gone that route.
One last thing- when I was 18 and had just moved out, I got my nose pierced. My mom's reaction was very hurtful and still stings. She made several comments about how ashamed she was when her friends saw me with my pierced nose. The fact that she would be ashamed and embarrassed over my appearance did not in any way make me think, "Oh, the piercing was a bad idea, I will remove it immediately!" (In fact, 12 years later I still have it), it just made me think she was judgmental and shallow. Our relationship was very rocky for the next few years after that, after all, if she was going to be that ashamed over a piercing I certainly did not think I could share anything else with her. I am just pointing this out so hopefully, if and when she comes home with a tattoo, you can hopefully have the RIGHT reaction ("tattoos aren't my fave, but I love you no matter what". Good luck!