Everyone has some great point for either side of this discussion. I too have my own opinion to this and it is as follows. (I don't have kids yet but its best to know what you plan to do)
I don't believe a Child can make an appropriate decision regarding art at a young age,(Take a kid to an art museum you will see what I mean) because tattoos are art, not just images. Would you not want your art that's adorned on your skin to be aesthetically pleasing? A person needs to be mature, cultured and educated in tattoos to be able to make an important decision that will last for the rest of their lives. It is my opinion that a child does not have these qualities. I am speaking out of experience, and I don't think I was ready at 18 to get one.(although I was not very educated in tattoos at that time in my life. I got that one covered up now) Even at 21 I got something that I am happy with the idea of but now at 31 I am thinking about what I can do to it to have it fixed, or should it just be a cover up as its in an area that is blocking bigger and better things. All my other tattoos have been carefully planned out and they look amazing!
Also when a child muscles are not fully developed an image may just become distorted as they grow into their prime. So no I would not let a child get tattooed under the age of 18.