What are the benefits of doing a PhD on a case study?
Case studies offer researchers the opportunities to dig deep into the rich details of the case.  In social science, this can frequently mean being able to answer "how" and "why" questions, use qualitative data, and generate rather than test theories.  I am not a physical scientist or engineer, but I imagine that similar principles apply:  the details of specific settings can be more informative than a more generic approach.

Case studies and quantitative/comparative studies are (in an ideal world) complementary to each other.  One tries to build up a deep understanding from looking carefully at a single (or smaller number of) cases, while the other seeks to establish generality, test theories, and (depending on the field) establish causation.

That said, in my experience (again, social science), the communities of scholars who work on case studies are frequently relatively disconnected from the communities of scholars who do statistical/quantitative work.  When choosing your career path, be sure to think about which community you will be happiest in.