When I first had my Touch, I suddenly became the new best friend of my buddy's 9-year-old boy. Of course all he really wanted were the games.
As for when is an appropriate age to own one, that really depends on the maturity level and risk tolerance that you're willing to accept.
Worst case, your kid will drop it, break it, or get it stolen. What will his/her response be? (a) Yell and scream and beg and sulk for you to buy another one, (b) accept that "them's the breaks" and be willing to move on, or (c) accept full responsibility including taking on chores or jobs to help pay for a replacement? And, knowing your kid, how long do you expect it to be before it happens again?
If it were my kid I would want them to be somewhere between (b) and (c). I'd encourage them to save up the money to buy it on their own, and learn about financial management, patience, self-discipline and property ownership while they were at it.
So my short answer would be, when they're capable of saving up and buying their own (whether they actually do so is the parent's call).