It seem certain that the intention in using "prochaine personne", which to me appears to be inspired by the English "the next guy", is to communicate something special, which unfortunately doesn't come through in French as there is no idiomaticity in this expression, except when talking about the succession of persons likely to have the same fate in a course of events. Maybe "Monsieur tout le monde" could be the idea, but I doubt it; nevertheless, it seems to me that there is no case for rendering this expression by means of a banal idiomatic substitute and that there is instead an author's problem, that of an author trying to put into words a new idea.
We get back into the domain of perfect idiomaticity if instead of "la prochaine personne" we use "la personne comme une autre", but I don't know whether that's what user Sylvester has really in mind.
The masculine gender can be used and then the reader has to conclude that this person is a male. What would be incongruous, for the least difficult to understand, would be an alternance in the text of feminine and masculine genders.