I'm not a huge fan of this sort of thing.
I personally think that in the age of entitlement that we are all currently living in and raising our kids in the concept of free money can be a hugely destructive one.
I think that a bank account for kids is a great thing and that depositing money into it that they got for birthdays and from their money box is a great way to encourage the concept of saving from a young age (and trips to the bank putting your coins into the counting machine is sooooooo much fun!)
As for the weekly deposits - there is no kiddies bank account that I am aware of that is paying interest even close to the level that we are paying on a mortgage or credit card or any other sort of loan so it makes more sense to pay it off those first. And then when the kids are 18 or whatever i will hopefully be in a better financial position that if they have worked towards something then I will be able to help out or give them a gift on my terms rather than having the enviable "but it's my money" brawl because there is a bank account with a lot of money ($10 a week for 18 years is almost $10000 before you add interest) that is in their name and they want to buy something that I don't think is appropriate and i havent worked hard to fund a ruby encrusted platinum navel ring
I like divvy's idea of matching their savings