Alex's question was how to test an API, not what makes a good API. Post-deployment monitoring seems critical to me. It's very hard, if not impossible, to know in advance what uses developers are going to want to put your API to once it goes live. Logging good requests tells you which features are popular, but logging bad requests and user support questions might give you more of an insight into aspects of the API that need to be improved. If you have the resources (and it's appropriate for your domain), generating a dashboard view of your service/API from the early days would allow you to track and prioritize improvements.
Pre-deployment, I think the main things you can try to measure are developer satisfaction and the "elegance" of code that's written against the API. Both very subjective things to test. Having a good set of primary use cases might help I suppose.