We discovered a few checks that we paid for customers in 2002, but never debited their accounts. Do we have any right to debit these items from their accounts now?

yes

Whether you have rights or not, I would look into the amounts and decided whether or not to do it. If it's a $25 check on a personal account that has been a good account, eat the check. If it's a $250 check on a business account that is a fee producing account and a good account, eat the check. If you decide to charge the check, I would do so in a deliberate manner rather than just have the charge "show up" in the account. A phone call to the accountant or customer would show some goodwill on the part of the bank and may lessen the strain to your call center down the road. My two cents.

I'll give a slightly different perspective...many years ago (late 80s), I banked at a S&L that was closed by the RTC. Four years after I closed my account, the RTC called me at work, and said that I had written a check in 1987 for $50, but the S&L had encoded in error for $25 and wanted their money. Thinking they must surely be joking, I asked for a copy. Sure enough, $50 encoded as $25. I contacted my trusty local banker, and inquired on whether or not this was legal. His response was "did you write the check intending to pay $50?" Of course, I said. Then you owe it, he told me.

Charge your customer.

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I would do so in a deliberate manner rather than just have the charge "show up" in the account. A phone call to the accountant or customer would show some goodwill on the part of the bank and may lessen the strain to your call center down the road.

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It's nice to see someone give directions for the High Road.