Safeguards verification of irradiated nuclear fuel assemblies in wet storage is frequently done by measuring the Cherenkov light in the surrounding water produced due to radioactive decays of fission products in the fuel. This paper accounts for the physical processes behind the Cherenkov light production caused by a single fuel rod in wet storage, and simulations are presented that investigate to what extent various properties of the rod affect the Cherenkov light production. The results show that the fuel properties have a noticeable effect on the Cherenkov light production, and thus that the prediction models for Cherenkov light production which are used in the safeguards verifications could potentially be improved by considering these properties. It is concluded that the dominating source of the Cherenkov light is gamma-ray interactions with electrons in the surrounding water. Electrons created from beta decay may also exit the fuel and produce Cherenkov light, and e.g. Y-90 was identified as a possible contributor to significant levels of the measurable Cherenkov light in long-cooled fuel. The results also show that the cylindrical, elongated fuel rod geometry results in a non-isotropic Cherenkov light production, and the light component parallel to the rod's axis exhibits a dependence on gamma-ray energy that differs from the total intensity, which is of importance since the typical safeguards measurement situation observes the vertical light component. It is also concluded that the radial distributions of the radiation sources in a fuel rod will affect the Cherenkov light production.