Oh, money. You hate it; you love it; you can't live without it. For the Kinkaid children, mo' money doesn't necessarily equal mo' problems, but it does add another thing for them to consider. As kids, they don't have a strong grasp on how much they need to live on the outside, but they are pretty quickly brought up to speed.
Jamie catches on right away, letting Claudia know that they don't have as much money as she thinks they do: "No, dear Lady Claudia. We have not the funds for taxis, buses, or subways. Shall we walk?" (5.12). Their crash course in money management represents their foray into the adult world, where they have to fend for themselves and save money for things as trivial (but necessary) as cracker packages.
For Claudia and Jamie, their money represents their independence. They are no longer subject to their parents, their allowance, and all those chores and rules. They can do whatever they want. But there's also a certain degree of responsibility that comes with managing their funds. And they have to accept that if they want to have the freedom that they desire.