Check with your local bank (not one of the big ones like BofA, Chase, Citi) but a local one to your city/state. A lot of times, they have kids savings accounts that pay 2% interest per year, no minimum amount but the only catch is that the person has to be under 18 years of age (hence, a "kids" savings account).
My local bank has that promotion and I'm like, ooh, too bad I'm not under 18!
Another idea, if your child is older, check into savings and checking accounts at Chase if you have a minimum amount. I keep on getting these deals in the credit card that if you open a checking account and have direct deposit, you can get $125-150 in 6 months (really, it's in 3 months but you need to keep the account open for 6 months). That's not a bad deal. Savings - if you have $10K - you can also open a Chase account and get $150 in 6 months. That's a potential of $275 - 300 in 6 months, which is pretty good!