ITunes and iPod Touch “Payment was declined”

The "official" way to fix, per Apple, it is to email Apple's iTunes support (iTunesStoreSupport@apple.com), inform them of the problem, and then wait for them to fix it for you on their end. The good news about this solution is that my email was answered very quickly, the support person knew what the issue was and how to fix it, and within a couple days of telling them about the problem they had fixed it and I am up and working again.

After they told me it was fixed, it did take almost exactly 24 hours to take effect, and in the email they told me "wait about 24 hours and try purchasing an app again."

At the time I was having this problem (while traveling abroad) I was in no mood to "wait it out" as other answers have suggested, but since it was handled so capably and promptly by iTunes Support, I wouldn't suggest that as a course of action. It is a known issue for them.

Gift cards are tied to iTunes accounts so if you want to let the store keep track of the different gift cards you will at least have to set up individual iTunes accounts for each of them (and yourself). Of course this also means that you have to switch the user in iTunes accordingly each time you want to make a purchase.

Thinking ahead, this approach is somewhat future-proof as well. In a few years your kids will switch to their own computer account with their individual iTunes installation etc. If they already have an individual iTunes account as well they can keep using their apps without having to bother you in case of upgrades etc.