(1) The generation of nanobubbles by electrolysis is an interesting method of using electrical energy to form bubble nuclei, effectively creating a multiphase system. For every process, the effectiveness of nanobubble generation by electrolysis depends on various process parameters that impact should be determined. (2) In this work, the electrolytic generation of hydrogen and oxygen bubbles was performed in a self-built setup, in which a Nafion membrane separated two chambers. The generation of bubbles of both gases was investigated using Box–Behnken experimental design. Three independent variables were salt concentration, current density, and electrolysis time, while the dependent variables were Sauter diameters of generated bubbles. An ANOVA analysis and multivariate regression were carried out to propose a statistical and power model of nanobubble size as a process parameter function. (3) The generation of bubbles of hydrogen and oxygen by electrolysis showed that different factors or their combinations determine their size. The results presented in this work proved to be complementary to previous works reported in the literature. (4) The Sauter diameter of bubbles increases with salt concentration and stays constant with increasing current density in investigated range. The proposed correlations allow the Sauter diameters of nanobubbles generated during electrolysis to be predicted.