The sum I've usually heard is about 50 cents per child's year of age, starting around 4 or 5. So a 4-year-old starts with $2/week, a 7-year-old gets $3.50 a week, and so on.
However, my husband and I also want to gradually, as our children get older, have them be responsible for paying for various things they want/need, and being able to sacrifice in some areas to save up money in others. So, for instance, at 12, our child will probably get a larger than usual allowance, but be expected to pay for her clothes (she'll have certain required amounts of clothes that she must have, including clothes for church), any books and movies, snacks, toys, etc. If she really wants $80 sneakers, she can choose to do that, but that might mean no candy for a while, or buying clothes at a consignment shop instead of new, etc. We haven't gone too far along that path yet since ours are still too young to really get money, but the one thing we've started is giving them a large enough allowance to pay for their own gymnastics and dance lessons. We'd like them to see how much money it costs to take gymnastics and dance and get an idea of how many LEGOs or bath toys they could have bought with that money, so that maybe they will appreciate the lessons a bit more. (No, they can't decide to use that money for toys.)
And we do something similar to kherbert, in that we also teach them to put money aside for tithe and charity right from the start. I think it makes it easier to maintain the habit if they start from the beginning. Right now I'm not worrying about "save" versus "spend," because my daughter saves for months to get LEGOs already (she wants a LEGO robotics set that will take her 2 years to save for), and I think expecting her to save for something more long-term than that (college) is just too abstract.