The effects of partial hydrogenation on the structure and electronic properties of boron nitride nanotubes are investigated via density functional theory calculations. We find that the structure of the nanotube may considerably deform depending on the exact locations of the hydrogen atoms adsorption. Furthermore, depending on the tube identity, diameter, and adsorbate density it is found that the bandgap can be gradually reduced as a function of the relative position of the hydrogen atoms and in some cases change its character from indirect to direct and vice versa upon hydrogen adsorption. Our findings indicate that partial hydrogenation may prove as a valuable and experimentally achievable route for controlling the electronic properties of boron nitride nanotubes.