With regard to the mind, I think that attempts at a purely positivistic understanding are not very promising. So I think that scientific research into the mind should, preferably, incorporate a significant 'internal' perspective. I explained this at greater length here.

Discounting law as a science, the only fields of research I know enough about to start to recommend specific methodologies are in social science (political science; political economy). Within these fields, the analytic narratives approach shows some promise in combining qualitative and quantitative research.

I do not think that this approach can be easily transplanted to anthropology because the set of issues dealt with in that field is just different. I also don't know how it would relate to mental research.

To my knowledge, it has never gotten past that stage to the actual work of science (like, in social science, formulating and isolating dependent and independent variables, testing them empirically, contrasting results with alternative hypotheses to show why a better explanation has been yielded...).