What would the pH of rainwater be if carbon dioxide in the atmosphere doubled?
"Cutting the $CO_2$ in half would yield about 5.9." 


(A valid approach because we expect reversibility in chemistry and physics.)

[pH of 5.9 for 200 ppm $CO_2$]

Atmospheric $CO_2$ of 200 would be about that of the last glacial maximum. So most of the difference between pure water at pH 7 and 5.9 would seem to be other gases dissolved in air. 

Side note: A drop to much below 200 ppm $CO_2$ would extinguish all life on Earth except those that live at geothermal vents at the bottom of the ocean, because they do not use $CO_2$.

If a linear relationship is assumed, then if there were 800 ppm of $CO_2$, rain water would have a pH of 5.0.  Though, if we assume the log relationship for concentrations that userLTK mentions in the comments, it would be a pH of 5.3.

Bouttes, N., D. Paillard, D. M. Roche, V. Brovkin, and L. Bopp (2011), Last Glacial Maximum CO2 and δ13C successfully reconciled, Geophys. Res. Lett., 38, L02705, doi:10.1029/2010GL044499.