I know, every parent's nightmare - "Mom, I want to get (a tattoo) or (my belly button pierced)." Take a deep breath and say, "OK, let's talk about it, there are lots of things to consider."
It may not make you feel better, but decorating bodies is not new. In fact, archaeologists have found carved figures from Stone Age people showing some sort of tattooing and figures back to 6000 B.C. include tattoos by most indigenous people. Piercing has also been common, but harder to document because the piercings will seal if the piercing object is removed.
Before you start the conversation, you might want to think about your own biases. Prior to the 1970s, tattoos were seen as "low life" and associated with gangs and criminals, but that changed when the baby boomers started experimenting with body art. Tattoos and piercing are pretty mainstream these days and many of the professional-looking people that you deal with every day are likely tattooed, yours truly included.
When you are ready to start the conversation, I suggest you thank your teen for talking with you about this decision and not doing something impulsive. Then, ask your teen why they want to get a tattoo, what design were they considering, and where they plan to put it. Things to consider are visibility - can they cover it up easily with clothing? Future - will it matter to a career, and permanency - they are hard to remove and the process will leave a scar.
Then there are the risks associated with tattooing and piercing. There is blood involved, so disease risk is important. State laws control body art studios and will require disposable needles and sterilization equipment to kill germs on jewelry. Reusing needles can spread hepatitis and HIV. There is also a risk that you will be allergic to the ink or that you will develop scars at the tattoo site. The iron oxide in some inks can also injure tissue during MRIs.
Specific to piercings, infection is a concern, especially for piercings around the mouth, which is full of bacteria, and specific to navels - clothes are going to rub it and healing takes a long time. It will not work to take jewelry out for school or work - the jewelry has to stay in for months during the healing, and the teen is going to have to soak the piercing with warm salt water several times a day for weeks, and whenever the piercing gets infected.
Another concern is health history. It is important to ask your doctor about any risks specific to you - some heart problems and conditions rule out any body art. If you and your teen decide that the tattoo or piercing may happen, it is really important to use a professional studio. Pick one and go visit it, ask to see the autoclave they use for sterilizing things, and make sure they do not use piercing guns (which cannot be sterilized) and that the needles are not re-used. In addition, check out the jewelry and make sure it is made from surgical grade stainless steel, 14-karat gold, or titanium. At a good studio there will be licenses hung on walls and the artists will be happy to explain the procedures.
So, now you have discussed the risks and done your homework. Lucky foryou, if you live in CA a person has to be 18 to get a tattoo, so no worries, they have to wait. If your teen is still interested, there are henna tattoos that last for weeks, so maybe that is an alternative, and might be a great way to see how people react, or to try different designs.
However, teens can get a piercing done, but it will require parental permission. That decision is up to you and your teen. There is magnetic jewelry that looks like a piercing that could be used first to see how it feels to have something on your nose or ear. Good luck!