I think that one of the important lessons that you can teach your child is how to manage money.  Recently, I signed up my 17 year old daughter for a debit card from PayPal.  PayPal offers a student program that allows teens to spend money with their own debit card.  You can control how much money is in the account at any given time.    Teens get an actual debit card they can use anywhere Debit MasterCard is accepted and it has their own name on it (they also have their own pin).  Anyone 13 and older can have an account and you get to see all of their transactions.  They can not spend more money than is on the account at any given time and you can set different alerts.

American Express offers a similar program called PASS.  PASS is a preloaded card - money can be added by the parent from their own American Express account, a bank account, or even by phone.  Teens can use the card anywhere American Express is excepted.  They can even use it at ATMs.  When you log on to the account, you can set alerts and see transactions.

Both of these options are great ways to teach your child to budget money.  For instance, if you provide them with $20 a week, part or all of this money can be loaded onto their card.  They can learn how to budget the money throughout the month, check their balances online and set up alerts to let them know when their card is low on funds.