Find a low-current safe dive location (SoFL or the Keys) and sign her up for a Discover Dive - sometimes called a Resort Course.
They'll spend some time in the pool learning the basics - then accompanied by an instructor will do a 1 on 1 ocean dive to a shallow depth. You could do it also.
After that you can pursue certification.
About the only question I would ask is, does she know how to swim?
Personally, I think 13 is grossly underage for this activity. A thirteen year old is not emotionally prepared to make life or death decisions that, while unlikely, can easily come about in this sport. I make a point not to dive on boats or with groups where children are present.
OP,
A 13-year-old's physiology and lack of psychological maturity, both, concern me, too. I'm not certain I would allow my own children to become certified at age 13 because of this concern.
Diversteve has a great suggestion, imho! If after your daughter does the discover scuba diver experience she is still motivated to pursue certification, you can approach things slowly.
My own girls are nine (twins) and 11. They have been taking weekly swimming lessons since age two. Two Christmases ago, they received masks and snorkels; this is their second summer using these in swimming pools. Last summer they breathed off scuba for the first time in my parents' backyard pool. Next month they will receive fins and weight belts (shhhhhh—don't tell them!) and they will again breathe off scuba when we visit my parents. The plan is to have them work with masks, fins, snorkel, and weight belts over the next year, and, if things work out, to take them to the Keys or to the Caribbean on a snorkeling vacation next summer.
Bottom line is, there is a lot your daughter can be doing to prepare for a scuba certification course, before actually enrolling in a scuba certification course. While she is spending a couple of years doing this prep work, she will be physically and emotionally/psychologically maturing, which imho is a good thing!
Safe Diving,
rx7diver
P.S. One other thing: If I weren't a reasonably advanced diver, I'm not sure I would be comfortable having my young children pursue scuba diving. I fully expect that, once they become certified (if they do become certified), then initially I, myself, will be the one diving with them for their safety.