There are many exciting open questions in behavioral market design, especially with regard to matching markets.
I mean for example the question of how people form preferences, e.g., over schools or university programs. This is pretty much a black box up to now, and it is often simply assumed that people have well-defined preferences when a centralized matching procedure is run. However, there is evidence that people search for information, that the process of discovering the preferences can be influenced by the market design, that people re-order their preference lists over time etc. (see for example the work by Narita and by Dwenger, Kuebler, Weizsaecker). Static mechanisms require people to submit their full preference list at once, but there are also dynamic mechanisms, e.g. in the sense of providing additional information as time elapses, requiring applicants only to provide their most preferred option within a certain set etc. Such mechanisms seem to gain importance, and have also only become feasible due to widespread internet access (see e.g. Bo and Hakimov, Gong and Liang).
A related topic that I believe deserves scrutiny is the question of what makes it simple to choose the optimal strategy in a (matching) mechanism and what makes it difficult. Strategy-proofness, that is, having a weakly dominant strategy to truthfully report one‘s preferences, does not seem to be sufficient according to evidence from actual matching markets and from experiments. Relatedly, how can market participants be convinced to report their preferences truthfully if this is in their best interest? Which type of advice works better than other? What type of information should be provided to the participants? (Some papers in this area are by Ding and Schotter, Guillen and Hakimov, as well as Pais and Pinter).