We present an ab initio three-dimensional precision calculation and analysis of high-order-harmonic generation (HHG) of the hydrogen molecular ion subject to intense elliptically polarized laser pulses by means of the time-dependent generalized pseudospectralmethod in two-center prolate spheroidal coordinates. The calculations are performed for the ground and first excited electronic states of H2+ at the equilibrium internuclear separation R = 2 a.u. as well as for the stretched molecule at R = 7 a.u. The spectral and temporal structures of the HHG signal are explored by means of the wavelet time-frequency analysis. Several aspects of ellipticity-dependent dynamical behaviors are uncovered. We found that the production of above-threshold harmonics for nonzero ellipticity is generally reduced, as compared with linearly polarized fields. However, below-threshold harmonics still appear quite strong except when the polarization plane is perpendicular to the molecular axis. Weak even harmonics are detected in the