The H i gas content is a key ingredient in galaxy evolution, the study of which has been limited to moderate cosmological distances for individual galaxies due to the weakness of the hyperfine H i 21 cm transition. Here we present a new approach that allows us to infer the H i gas mass M (HI) of individual galaxies up to z approximate to 6, based on a direct measurement of the [C ii]-to-H i conversion factor in star-forming galaxies at z greater than or similar to 2 using gamma-ray burst afterglows. By compiling recent [C ii]-158 mu m emission line measurements we quantify the evolution of the H i content in galaxies through cosmic time. We find that M (HI) starts to exceed the stellar mass M at z greater than or similar to 1, and increases as a function of redshift. The H i fraction of the total baryonic mass increases from around 20% at z = 0 to about 60% at z similar to 6. We further uncover a universal relation between the H i gas fraction M (HI)/M and the gas-phase metallicity, which seems to hold from z approximate to 6 to z = 0. The majority of galaxies at z > 2 are observed to have H i depletion times, t (dep,HI) = M (HI)/SFR, less than approximate to 2 Gyr, substantially shorter than for z similar to 0 galaxies. Finally, we use the [C ii]-to-H i conversion factor to determine the cosmic mass density of H i in galaxies, rho (HI), at three distinct epochs: z approximate to 0, z approximate to 2, and z similar to 4-6. These measurements are consistent with previous estimates based on 21 cm H i observations in the local universe and with damped Ly alpha absorbers (DLAs) at z greater than or similar to 2, suggesting an overall decrease by a factor of approximate to 5 in rho (HI)(z) from the end of the reionization epoch to the present.