The main factor to consider is the child's mental and physical maturity as compared to the demands the art will make on him or her. Martial arts training (indeed, any physical activity) should be geared toward enhancing a child's physical development and mental maturity.
Thus, though it should go without saying, many hard-core competitive striking martial arts are not really physically appropriate for children until they are well into their high school years. By this, I mean arts such as boxing, Muay Thai and Isshin-ryu karate - which advocate contact sparring as a core component of their training. There are those that say that you can train these arts without the contact component - but then you end up watering down the essence of the art, so what's the point? Better to find a style that your child can practice full-on with minimal fear of incurring broken bones, facial injuries and blows to the head that could affect his or her learning ability. Such styles also place stresses on the joints that are inappropriate for young bodies already struggling with the considerable demands of growth spurts.
Arts that encourage flexibility and balance are wonderful for young children, however. These are attributes that are somewhat easier to develop during childhood and stay with a person throughout life without minimal maintenance. Thus, Korean arts such as Tang Soo Do and grappling arts such as judo, wrestling or Brazilian jujutsu are great for kids. Children are generally ready to study these arts around the time they begin elementary school.
The main factor for whether a child is ready to train a particular art lies in his or her mental capabilities, however. On one hand, the art has to be interesting and fun for a child. Thus, the more philosophical and meditative arts, such as Taijiquan and Aikido, might bore most children (and even some young adults). While there are exceptions, it is rare to find anyone with the requisite interest and patience younger than 18.
