Have a look back at the agreements/contract/communication. Make sure that what you produced falls within the boundaries of what you understood was being asked for. It is a tricky area particularly for freelancers or small businesses. Often the clients expect the world for chump change. That means you have to be able to show that you have done what the client asked and it also means you need to have really clear terms of business. ie. Work covers job A and is expected to take 4 hours, if this time is exceeded the chargeable rate is $X/hour. Always build something like that in. It means you can't get screwed out of work like this.
Sometimes you will get it wrong and under-estimate the job. That's just business and sometimes you will have to weigh up whether being paid is better than not. That might mean you take a hit on an hour or two of your time. But you still get some money versus none.
Just this week I had a client present quite a different picture of expectations than we understood. It was a website in flash that he wanted to play on an iPhone. Due to the width is was going to be illegible to keep its format so I kept to what I thought was a good standard. He turned around and said no I want it exactly the same as the flash version, I don't care if the clients can't read it! Ok...
We were faced with the, is this going to end badly and do we pull the plug now and offer to drop the job for zero money. In the end we gave him 3 options. Walk away and pay nothing and put it down to miscommunication. Tweak what we had produced to make it more like what he wanted. Or take what we've done and employ another contractor to tweak it.
In the end it was really just a personality quirk combined with his lack of understanding about what is/isn't possible on the platform. We made a few adjustments and he was able to say "that's EXACTLY what I wanted", bill paid happily.
It highlights the importance of good communication and clear guidelines. You want to cover yourself from those that might seek to take advantage, but even if the client hates your design if you can prove that you have reasonably fulfilled the contract it is difficult for them to avoid paying. Just remember that business is not all about the money. Reputation and client satisfaction go a long long way.
In your case I would try and establish what the core issue is. If you can show that you have done the work you might reduce the amount the client has to pay but tread carefully.
I was wondering, How do you guys respond to chargebacks or Refunds, when a customer is not happy with your work?
That's simple for me; I don't give refunds at all, except for when I turn ill and cannot complete a project according to what had been initially agreed on.
Before a project starts, you need to ask the client what he wants. You should make sure you get an idea of what it is he definitely doesn't want. I find it's always a good idea to let the prospective client give you examples of work he considers well executed.
The client knows your style (portfolio), and the client has given you samples of work he considers great. If you keep that in mind while designing, then you should be in a much more confortable position.
The most important thing, however, is to take everything you've discussed with your client and translate that into a contract that you and he need to sign. I don't do work without a proper contract anymore, and in this contract I have a clause where I explicitly state that I do not give refunds. However, due to having made it a habit to communicate with my clients in a very, perhaps overly detailed manner, so that we both know exactly what to expect of one another. So far, this has worked the best for me and since I've gone this route, my projects have turned out much better than before.