I started playing tennis at 10-years-old. My brother Paul started when he was five. I think it helped Paul to start at an earlier age and his playing achievements far surpassed mine.
However, that is not always the determining factor and it is important to approach each child on a case by case basis. I have taught quite a few 3 and 4 year olds who are capable of being on the court for an hour and enjoy it.
On the other hand, I have worked with kids 10, 11, or 12 years old (or older) that could not focus for five minutes in a row. The most important factor in my opinion is the child's ability to stay attentive. If they are all over the place with their attention span, a shorter period of time may work or it might make sense to wait and try again in a few months or even a year from their first attempt.
Again, kids are all different and there is not one specific age that I would universally say, is a good age to start. Just give it a try when they show some interest and often a good pro will be able to engage them and allow even the very young players to have fun and ask for more.
Steve Annacone, USPTA Elite Pro, is the Director of Annacone Tennis (http://www.annaconetennis.com) and MyHamptonsPro (http://www.myhamptonspro.com). For details on lessons and clinics throughout the Hamptons contact firstname.lastname@example.org or email@example.com or call 865-300-7323