pH of rain with and without thunderstorm
Though I've never heard of that statement before, it's possible that nitrogen oxides formed by electrostatic discharge (not only lightning, but even glow discharge) dissolve in the rainwater, producing nitrous and nitric acids. 

There is a 1997 citation describing comparison of pH in rain accompanied (or not) by lightning.

That said, presence of fly-ash from coal burning, sulfur oxides from volcanic activity and carbon dioxide from natural as well as anthropogenic sources probably have greater influence on pH.