How should you compensate a customer for a service or product failure?

It depends. And that variability, in fact, is what's most important. Customers have diverse values and preferences- so your people who placate disgruntled customers need to be given enormous discretion. Still, there are principles that apply:

• Most customers understand that things can and will go wrong. What they do not understand, accept, or find interesting are excuses. For example, they don't care about your org chart. You mentioning that a problem originated in a different department is of no interest to them.
• Don't panic. Customers' sense of trust and camaraderie increases after a problem is successfully resolved, compared to if you had never had the problem in the first place. This makes sense, since you now have a shared experience. You have solved something by working closely together.
• Avoid assuming you know what solution a customer wants or "should" want.Ask. And if a customer makes a request that sounds extreme or absurd, don't rush to dismiss it. Even if it seems on its face impossible, there may be a creative way to make the requested solution, or something a lot like it, happen.
• Don't strive for "fairness" or "justice". Our archetypal doting Italian mama doesn't investigate whether her bambino obeyed the sidewalk speed limit before comforting him, and a customer's warm feelings for a company aren't about fairness. They're about being treated especially well.
• Learn from customer issues, but don't use them as an opportunity to discipline or train your staff in front of your customer. This may sound obvious, but it happens quite often. Watch out for this flaw, special when you're under stress.
• Don't imagine you are doing something special for a customer by making things how they should have been in the first place. The chance to get it right the first time? It's gone. So re-creating how things should have been is just a first step. You need to then give the customer something extra. Mam bandages a knee and offers a lollipop. If you aren't sure which "extra" to offer a particular customer, just make it clear you want to offer something. If the customer doesn't like red lollipops or doesn't eat sugar, she will let you know. Then you can decide together on a different treat.
• Keep in mind the lifetime value of a loyal customer. A loyal customer is likely worth a small fortune to your company when considered over a decade or two of regular purchases. Research shows that the lifetime value of a loyal customer to be up to $100,000 and occasionally more. It is well worth figuring out that number and keeping it in mind if you ever feel that temptation to quarrel with a customer over, say, an overnight shipping bill.