Voice of cautious reason: Tread carefully in this decision.
I helped my girls get a credit card and a checking account when they were each 17 -- the summer before their senior year in high school. That gave them a year of living at home /using those items under my supervision before they started college. The credit cards only had a paltry $300 limit. Why? Because -- if they didn't follow my advice -- I didn't want them to have the ability to dig themselves in too deeply. They learned how to use "invisible money" well, and neither has ever had a problem with credit cards or checks -- but you're talking about a child two years younger. She'll need more supervision.
Done well, your proposal can help her learn to manage money; done poorly, it can hurt you both and cost more than the $50 gift card.
If she's an authorized user on your account, will she have full access to your entire credit balance?
And, yes, definitely check to see if an annual fee is part of this card.