Your 93 probably does not support plugging in a code reader to pull a code for the ABS system. You should be able to jump a couple of pins in the diag box and count the flashes of the ABS light to get the code out of the ABS ECM. I just sold the ABS ECM out of a 95 M, I still have the ABS hydraulic control unit but I hope you do not need it. The rain won't have any effect on the ABS system (or shouldn't). My guess would be a wheel speed sensor. I do have three of those left, one sold. On the Miata.net web site you should be able to find the info to tell you what pins to jump, if not let me know, I imagine I can dig it up for you.
thanks, Craig