Is there any biblical basis for giving a child an "allowance" -- a fixed amount of money, typically every week or month? Indeed there is.
"Whoever is faithful in a very little is faithful also in much; and whoever is dishonest in a very little is dishonest also in much. If then you have not been faithful with the dishonest wealth, who will entrust to you the true riches?" (Luke 16:10-11)
The Lord teaches that character is built and trust is gained through small victories in small things. We teach our children to do three things with their money: spend some, save some, and give some away. Even a child as young as 5 or 6 can begin to learn about money. And the sooner, the better!
For more practicals on your children's allowance money, see The Quiver: Christian Parenting in a Non-Christian World.