Lol, tell her you'll get a matching one. That might put her off.
Seriously, though, I understand what you mean -- the idea of our "perfect" children wilfully marking their bodies is uncomfortable. But you're right, it's pretty culturally normal to get a tattoo. In fact it's probably a rite of passage. Also, I'm sure you did some things to your own body that your parents weren't totally happy about. (Or maybe you didn't, but I know I sure did).
As far as the actual tattoo she plans to get, and where she wants to get it -- well, it's going to be pretty inconspicuous. There are worse things. Think of it like a scar. Scars aren't always pretty, but they do give you character. It doesn't even matter what you get. For those of you who feel that a tattoo should reflect some deep and meaningful truth, forgive me. I don't agree. The act of marking your body is what matters. This is important, because she will almost certainly not care about the tattoo she gets now when she's forty. I mean, she may not hate it, but she won't think it's particularly clever anymore. What's meaningful now, at this young age, may not be later. (I think this is different if you get a tattoo when you're thirty; your tastes then might have more staying power).
Anyway, have that cup of tea -- I'm there with you in spirit! -- and then chalk it up to one of those little things we, as mothers, have to let go.