Our daughter, 17, has a savings account. She is also an officer in our company and on my payroll. I know Chase banking has a student account with no fees until they are out of college or 26 yrs old. She also has a savings account at BOA. We have accounts set up at both banking locations.
She ends up with a nice stash of cash and she does not carry it with her. She is a popular babysittter and makes an easy $100 per weekend cash. Cash is so easy to lose, even for a responsible teen. I have "rule" for her to have a least $20 in her wallet at all times and she has her company credit card. If she is going shopping for clothes, etc, she will take her cash.
We pay for ALL school related expenses including her food as well as her car expenses which is only gas. She pays for her clothing, fun things, etc.
She has a set limit on the credit card for things such as school lunch or out with friends lunches, occasional clothing, etc. She has never abused the card but we expected that she would not because she knows debt is evil and we live debt free. She knows the deal that the card is always paid in full before statements are even cut, there is never ever a balance or fee incurred with credit card or banking.
The card is helpful because she is a cheerleader and often away at games, etc and we don't want her in a position to be without any access to funds.
People may not agree with me but I do believe children model the behavior of their parents. If parents are witnessed splurging on things or abusing credit cards or money, etc the child is more likely to as well.
In our house, we have taught delayed gratification from day 1 and not owing anybody anything. She shadows me with accounting and finances with our company and she gets it... We run personal finances much like our company..... get a bill/invoice in the mail... you pay it that day so it is gone and taken care of.
I think your idea is just fine for a responsible 16 yr old. She needs access to funds when she is away from home. Set guidelines, give some rope and see how she does then loosen the rope as she gets better with her financial decisions.
Good luck!