Is there such a thing as an interdisciplinary PhD, where the student chooses the fields, courses, etc.? If so, which U.S. universities offer such a thing?
For a while I was a graduate student in the math department of Portland State University. At Portland State, the Ph.D. program in math requires students to have an "allied area" (field of study other than mathematics). The student's thesis must be related to this allied area, the student must take about 25% percent of their courses in the allied area, and the student must pass a qualifying exam in the allied area. Thus the whole program is designed to be interdisciplinary from the get-go.
(Side note: The actual name of degree is "Ph.D. in Mathematical Sciences", and it definitely differs from the traditional math Ph.D.)
In general, every Ph.D. is officially within some formal department, or other program, and a Ph.D. student will need to do at least whatever courses that program requires. Some such programs, however, are extremely interdisciplinary by their nature: some nice examples are MIT's Engineering Systems Division and Media Lab, where participants have the opportunity to take a wide variety of different courses connected to different disciplines. Every program has some (often relatively loose) expectations about what a student will do, however, so you're unlikely to find anything where you just get to pick whatever you feel like.
Beyond a certain point, however, a Ph.D. is not about coursework. Once you start to focus on your research, then you can do anything that you and your advisor agree is appropriate...