As for the age, that is completely up to you. I, personally, could never pierce a child's ear without them saying they want too. Some grow up to just not like it and that is not my body to make that decision.
As for the place, contradictory to what many are saying, please take her/him to a reputable (which does not always mean cheapest!) tattoo shop in your area and ask the piercing professional there to do it. There are a plethora of risks associated with the guns Claire's, Walmart, or any local piercing pagoda has which include
-Risk of infection as they cannot clean the 'guns.' They rub alcohol on them, but that is nothing compared to what a piercing professional does. They put needles in an autoclave which sterilizes the instrument COMPLETELY. The infections from piercing guns can go anywhere from a simple easy fix, to MRSA, and even HIV. The stats vary depending on how highlu populated the area is, but that still isn't a chance I would ever take.
-Untrained young adults. These people often practice on teddy bears or something on the like. A professional often apprentices anywhere from 2-4 years under another professional. 2 weeks on a teddy bear, or 2-4 years working with a professional on real people?
-Healing is much easier with a needle piercing than a gun. The needle is sharper, hurts less, and heals much easier. They also use 316L Surgical Stainless Steel or Titanium, which means you also avoid the possiblity of your child being allergic to nickle because sometimes 'nickle free' jewelry still have nickle in the saudering of the post and the jewelry. My mother can't wear even high grade expensive earrings because of that. I have 6 piercings and the two that I had done with a gun years and year ago I had problems with til about 4 years ago when I started stretching them.
If you do choose to go to a professional, ask the piercer to use a labret (Pronounced la-bret) stud. They are the studs used for lip piercings and are much more comfortable because they are not a sharp 'starter' earring. You can get cute fronts as well, so she/he won't be stuck with 'boring' jewelry. The gauge to get them pierced is 18 and don't use anything but sea salt soaks, such as H2Ocean. The commercial 'ear cleaning' solutions have hydrogen peroxide in them which can be very detrimental for piercings. Do not touch or turn the jewelry/piercing either. After a 6 week wait, you can change the jewelry and wear regular earrings.
If you have any questions I would be glad to help, body modifications is a passion of mine and I am studying myself to be a piercing professional.