The reason why the vast majority of people are so poorly educated about finances is there's a taboo against talking about money, so I think the best thing you can do is talk to your kids about the family finances. Parents are usually okay about the basic lessons on savings when kids are very young, but I think we fall down when they become teenagers. Start talking to them about specifics, like income, bills, loans, investments, retirement accounts, inheritance, trusts and so on. Don't be afraid to show them the real accounts and numbers.
Of course if they're not born yet, that's a little premature. In that case, you might want to consider something like a 529 account. Always look at the one in your state first, because you might get some tax breaks, but they're not geofenced. If your state 529 sucks, just find a top-rated program in another state. It's a good way to put away money for their future education.
Another thing you might also consider, as they get older, is retirement account matching. Offer to match 100% up to some level of the money they put into an IRA (the gift limits are a couple times the IRA contribution limit, so no problems there). While IRAs require earned income, $1,000 from mowing lawns or babysitting counts. If you match it all, then they'll still have a $1,000 to spend, but also have an early start on retirement. I think that's a better gift than some money in an envelope -- 50 years at 8% would turn every $1 into $47. But the real benefit is the experience -- after all, when people are asked why they didn't invest earlier, the main excuse they give is they didn't know where to start.
