My son has had a debit card, linked to his checking account (but with my name on the card) since he was 9...yes, 9 and it was the most brilliant thing I ever did.
Money was a magical thing before then. I would never have cash on allowance day so I kept a mental tally of what I owed him. He would never have his allowance on him when at the store so we would "deduct it from his allowance" but we know how good we were at that. So, money became this never ending thing and we never knew who was up or down. He felt he just had to buy something every time we went out.
Once I opened his checking account and got him his own debit card, money became real. I schedule the kids direct deposit allowance so I never forget to pay them. When he wants to buy something, we check his balance. Suddenly, money was a real thing and he never wanted to buy everything in the store anymore. He became very conscious of his money.
He is now 16 and saving for a car (has a job). When we go out of town and leave him alone, he has a debit card to order pizza. He has the card linked to his xbox account and such.
I am not sure the age to get him his own card in his name but at age 9 it had to be in my name. I will have to get around to changing that one day.
My daughter (age 14), on the other hand does not have a debit card yet because a) she doesn't spend money and b) isn't really out in the world yet and c) I just have not gotten around to it yet.