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savingadvice
I'm starting my saving early (at 4 months!) and one offer for kids I came across was for Affinity Bank (http://www.affinitybank.com/pb/savingsKids.asp). They offer a "Kids Only Savings Account" that pays 10% APY interest rate on all balances up to and including $500. That rate is over 250% better than even the highest savings accounts are currently offering.

The minimum deposit on the account to begin the account is $1.00. Balances that exceed $500 will be paid the bank's regular savings rates. Children that open an account also receive a free Piggy Bank!

The Kids Only account is restricted to children under 16 years of age and their parents must have or open an account with Affinity Bank (they do offer a free checking account with a minimum $100 balance). Certain other restrictions and transaction limits apply.
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4 month old baby already saving for college -Financial Baby Steps (http://financialbabysteps.blogspot.com)

enoughwealth
You could also shop around for a fee-free account that offers a joining bonus - for example, my son opened an online account (with me as "member-get-member" referrer) and got a $20 bonus on a $5 deposit.

ps. Regarding the "Kids Only Savings Account" - The 10% APY on the first $500 is only good if you plan on keeping $500 or less in the account. If you had a large balance (say $2000), then having $1500 of this earning just the bank's regular savings rate (0.01% ?) would make it a bad choice. Hopefully you could link it to, say, an ING online account and shift any balance above $500 into the linked online account.

Regards
http://enoughwealth.blogspot.com

peacefulbell
compounding interests is really powerful, great to start young!