We investigate the effect of hydrogen coverage on the optical conductivity of single-side hydrogenated graphene from first-principles calculations. To account for different degrees of uniform hydrogen coverage we calculate the complex optical conductivity for graphene supercells of various sizes, each containing a single additional hydrogen atom. We use the linearized augmented plane wave method, as implemented in the WIEN2K density functional theory code, to show that the hydrogen coverage strongly influences the complex optical conductivity and thus the optical properties, such as absorption, of hydrogenated graphene. We find that the optical conductivity of graphene in the infrared, visible, and ultraviolet range has different characteristic features depending on the degree of hydrogen coverage. This opens up new possibilities to tailor the optical properties of graphene by reversible hydrogenation, and to determine the hydrogen coverage of hydrogenated graphene samples in the experiment by contact-free optical absorption measurements.