This paper proposes the use of solar-sail technology currently under development at NASA Langley Research Center for a CubeSat rendezvous mission with asteroid 2016 HO₃, a quasi-satellite of Earth. Time-optimal trajectories are sought for within a 2022-2023 launch window, starting from an assumed launcher ejection condition in the Earth-Moon system. The optimal control problem is solved through a particular implementation of a direct pseudo-spectral method for which initial guesses are generated through a relatively simple and straightforward genetic algorithm search on the optimal launch date and sail attitude. The results show that the trajectories take 2.16-4.21 years to complete, depending on the assumed solar-sail reflectance model and solar-sail technology. To assess the performance of solar-sail propulsion for this mission, the trajectory is also designed assuming the use of solar electric propulsion. The resulting fuel-optimal trajectories take longer to complete than the solar-sail trajectories and require a propellant consumption that exceeds the expected propellant capacity onboard the CubeSat. This comparison demonstrates the superior performance of solar-sail technology for this mission.