We report here on studies of reorientation of human red blood cells (RBCs) in an optical trap. We have measured the time required, , for the plane of the RBC entering the optical trap to undergo a 90-deg rotation to acquire an edge on orientation with respect to the beam direction. This has been studied as a function of laser power, , at the trap center. The variation of with increasing shows an initial sharp decrease followed by a much smaller rate of further decrease. We find that this experimentally measured variation is not in complete agreement with the variation predicted by a theoretical model where the RBC is treated as a perfectly rigid circular disk-like body. We argue that this deviation arises due to deformation of the RBC. We further reason that this feature is dominated by the elastic behavior of the RBC membrane. We compare the studies carried out on normal RBCs with RBCs where varying conditions of membrane stiffness are expected. We propose that the value of energy used for maximum deformation possible during a reorientation process is an indicator of the membrane elasticity of the system under study.