Natural rain water is slightly acidic (pH = 5.6) since there is carbon dioxide in the atmosphere that dissolves to make carbonic acid.
What would be the pH of rainwater if carbon dioxide in the atmosphere was doubled to 800 ppm?
"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.