... give it to them.
A true respectable company is a company that customers will come back to over and over again. If, as a customer, I have to argue my point, I am going to be left with a sour taste in my mouth and will inevitably tell my friends about my awful experience.
Whether it's being billed unexpectedly (we see this a lot with re-bill programs), erroneously (PayPal isn't flawless), or just billed for something that wasn't what was expected, the customer is always right. Do the right thing, explain what happened, and offer something for free to this customer for the troubles.
Most likely, the customer doesn't hate you. They might not even be too upset at this point. Most rational people are forgiving, and it's those people that, although you're refunding them now, will come back later on to hire your company again in the future.
Remember, it's ALWAYS about the customer experience.