One method that my significant other's family used when she was growing up was matching any funds that she saved. That is, you give your child an allowance (let us say $2 a week), and give her a choice: tell her she can spend the money however she wants. She can go out that week and spent $2 on candy or whatever and enjoy it. However, you promise your child that if she decides to save the money, you will put the money in a savings account and match her $2 with an additional $2 (or whatever you think is appropriate), and therefore deposit $4 into an account in her name. I like full matching because she can more easily see how fast her money grows if she saves it. She can more easily appreciate the benefits of delayed gratification that will actually increase her future income. She will more easily understand that by spending her allowance immediately she gets $104 dollars, but by saving it at year end (excluding interest, which is negligible these days), she will have $208.
Now, this savings-matching program is exactly what many employers do with 401k savings programs to encourage grown adults to save (even though everyone should save for retirement anyway).
I should also say this is not foolproof: my significant other saved all of her allowance, but her sister spent all of her allowance every week despite the great incentive! But again this isn't different from real life: many employees decide not to save as much as their employers will match dollar for dollar under their 401k programs even though their employers offer them FREE MONEY to do so!!! Sorry to shout, but it is insane to me that people will turn down FREE MONEY.