To add to the last two posts:
Twenty-first century financial market design is a great area for dissertation research. There are lots of incremental changes in rules and practice, and many opportunities to better understand those changes and to propose improvements. One example of which I am aware is Jun Aoyagi, a Ph.D. student at UC Berkeley, who has a paper on Strategic Speed Choice when the exchange imposes small delays. There surely are others already working on related topics, but I conjecture there is additional room for dozens of dissertations in this field.
The earlier suggestion about designing markets for personal data also strikes me as important and promising. The book by Posner and Weyl is useful background, and it frames the question nicely, but it doesn't really tackle market design. Individual users are numerous and tiny relative to giant firms like Amazon, Google and Facebook on the other side of the market. I wonder whether there is a role for intermediaries to sign up large numbers of users and sell the bundled data at a competitive price. In any case, there seems to be room here for basic theoretical modeling, for specific applications and for everything in between.