You know, it's probably just me being over protective, but I don't think I'd let my 13 year old get the belly button done. Just because as akmom said, why does she want it? 13 is young to be showing off their belly, you can't do that at school anyway, and it seems like just a temptation to dress inappropriately and whatnot. Plus I would be making it a sexual thing in my head and it would freak me out (as in, she wants guys to see it when she's naked etc, I know 13 is so young to say that about but not these days! I see and hear stuff all the time about 13 year olds being sexually active and it freaks me out!!). I don't think I would allow any facial or body piercings on either gender until 16, same with tattoos, and that's mainly because I don't want them to make any permanent marks on their bodies and especially faces until they are old enough to really know that they want them to stay there, you know? And to make good decisions on WHAT they want on their bodies forever. I say piercings too, because they do leave marks where you put them. I used to have an eyebrow ring, and I haven't worn it in a couple of years but the mark is still there and I wish it wasn't. I might just get it put back in for the hell of it though haha.
DH used to have his septum done, and that hole doesn't really show, but you can see where he used to have his lip and eyebrow done. Also his ears were gaged quite large and they are stuck that way as well. He would still be wearing all of these, but he isn't allowed to at work so he just doesn't bother anymore. If given the opportunity though, I know he'd put them all back in haha. He also has his belly button done.
I also wouldn't allow ears to be gaged until at least 15 or 16 for the same reasons (DH's ears as an example), but regular sized ear piercings of any kind I am fine with on either gender at any age. My 6 year old has one of his ears pierced. My 2 year old doesn't have his done, but if he asks for it I'd gladly let him. Any type of ear piercing, just not gaged.
My ears are pierced twice on the lobes (I kind of want a third set, actually), and once on my left cartilage. I still wear earrings in all of those. I only have one tattoo, but I do want more. DH has... idk, he's not home for me to count them lol, but I'd say maybe 15 or 16? And I tattoo as a hobby at home, so I think maybe 13 of them are from me. He has them on his arms, legs, hands, and neck. They don't cause him any trouble at work.
I just want my kids to wait for the more permanent stuff, because at 13 I don't think they are very capable of making very good decisions about permanent things (that's not so much about the belly button ring, with that like I said I just wonder how appropriate that is for a 13 year old).