Since this post was migrated from Parenting, my reply was in the context where it appeared to be misrepresenting facts to make a point. I've edited it to be more concise to my main point.
In my opinion, the best way to save for your childs future is to get rid of as much of your own debt as possible. Starting today.
For the average American, a car is 6-10%. Most people have at least a couple credit cards, ranging from 10-25% (no crap). College loans can be all over the map (5-15%) as can be signature (8-15%) or secured bank loans (4-8%).
Try to stop living within your credit and live within your means. Yeah it will suck to not go to movies or shop for cute things at Kohl's, but only today.
First, incur no more debt. Then, the easiest way I found to pay things off is to use your tax returns and reduce your cable service (both potentially $Ks per year) to pay off a big debt like a car or student loan. You just gave yourself an immediate raise of whatever your payment is. If you think long term (we're talking about long-term savings for a childs college) there are things you can do to pay off debt and save money without having to take up a 2nd job... but you have to think in terms of years, not months.
Is this kind of thing pie in the sky? Yes and no, but it takes a plan and diligence. For example, we have no TV service (internet only service redirected an additional $100/mo to the wifes lone credit card) and we used '12 taxes to pay off the last 4k on the car. We did the same thing on our van last year.
It takes willpower to not cheat, but that's only really necessary for the first year-ish... well before that point you'll be used to the Atkins Diet on your wallet and will have no desire to cheat. It doesn't really hurt your quality of life (do you really NEED 5 HBO channels?) and it sets everyone up for success down the line.
The moral of the story is that by paying down your debt today, you're taking steps to reduce long haul expenditures. A stable household economy is a tremendous foundation for raising children and can set you up to be more able to deal with the costs of higher ed.