Doesn't matter to me, I'd rather a girl not have one though...It will tempt her into showing it off which would make her appear as a slut or something, and I'd rather keep my girls body to myself...but its all in why and how you use it.
When you get one you really have to where a short top to show it off, thereby exposing yourself in the process, it draws attention to your body, the ring itself may not be trashy, but the cloths or the lack thereof maybe considered trashy. From a guys point of view doenst really matter, if you like the girl its like a fashion statement, like a tattoo, can be removed or covered up not a real big issue. Kinda prosaic...
I don't think they're trashy, I just don't particularly like them. I like body mods in general, but navel piercings are among my least favorite. They're the 'every-girl-hits-adolescence-and-wants/gets-one' piercing.
He probably thinks they're trashy because they're typically under clothes. So with that logic, what's the point of having one if you don't intend to let people see it? It's probably the 'showing off' that he finds distasteful, and not something he wants for his daughter, especially as a minor. One, he doesn't want people seeing anything that should be UNDER your clothes, and two, he doesn't want you feeling encouraged to SHOW people what's under your clothes. It's a dad thing. And I agree with Yengesi about the dad's house, dad's rules thing.
Why not just wait? If you still want it by the time you're 18, then you have every right to get one. Keep in mind that they're among the most difficult piercings to heal, take foreeever to heal completely even under perfect circumstances, and tend to have a lot of problems. They're hard to keep clean, easy to get infected, skin often rejects the piercing, and they get in the way of everything during healing.
Personally, I've always liked them. I have one on top and bottom. But I like piercings and tattoos, and tend to get them where I can hide them so I still look professional, so belly button rings were perfect for me.