Assuming you are talking about a house cat and not an outdoor farm cat, the answer to this question depends on a few factors. The first is whether or not the child has allergies. If the child is allergic to cats but really wants one, there are now hypoallergenic breeds available that may work (although some people may still be allergic to these). Secondly, consider whether the child is responsible enough to handle feeding, litter box duty, and general caretaking. A good rule of thumb is that if the child can read chapter books and do simple math, they are probably ready for a pet. Finally, think about whether or not you as the parent are prepared to take on most of the caretaking responsibilities if necessary - remember, a pet is for life!
So when it comes down to it, there is no definitive answer as to what age a child should get a cat. It depends on individual circumstances and readiness. However, if you do decide that your child is ready for a feline friend, be sure to choose a hypoallergenic breed and help them understand what caring for a pet entails.