I reckon 15-16 would be ok but maybe not 13. Depends on how mature she is I suppose. There’s also the risk that her friends will follow and do the same, which could be tricky if they’re a bit less mature. Although it’s obviously not your issue as they should ask their parents first. Just something that I was pondering.
Haha. I had mine at 16 and didn’t tell my mum but I already had my nose, eyebrow and tongue pierced so it didn’t really make a difference. The only thing I will say that getting your navel pierced hurts like hell - at least it did with mine and I was quite experienced with piercings. I’d also recommend a bar as they’re less likely to cause hassle. I had a ring which was sore as it kept moving around so I took it out and the hole healed immediately, so in order to get a bar I had to get it re-pierced which again hurt like hell. 
My only stretch mark from my pregnancy is on the bit where I used to have the belly bar. It stretched and now it has a tiny scar on it. Doesn’t bother me at all but something to consider perhaps.