Is my electrolysis emitting compounds from additional reactions?
TL;DR: Your suggestion that the problem is "simply bits of $\ce{NaOH}$ being carried into the air by the splashing bubbles" is very likely correct.  

You have a very simple system, meaning a minimal number of variables to consider in order to eliminate other possibilities. For example, if you were using $\ce{NaCl}$ as the electrolyte, then chlorine gas could be the irritant. But with $\ce{NaOH}$ as the electrolyte there really is no side reaction that could produce such an irritant.  

On the other hand, $\ce{NaOH}$ itself can irritate sensitive tissues like mucous membranes, and even skin for that matter. According to one of your comments, you are using a $\ce{NaOH}$ strength of a bit over 1 molar. Droplets of this solution will eventually be neutralized by $\ce{CO2}$ in the air, but at that strength it should stay around long enough to elicit the symptoms you described.