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We present a theory of phonon-drag thermopower, $S_{xx}^g$, in MoS$_2$ monolayer at a low-temperature regime in the presence of a quantizing magnetic field $B$. Our calculations for $S_{xx}^g$ consider the electron-acoustic phonon interaction via deformation potential (DP) and piezoelectric (PE) couplings for longitudinal (LA) and transverse (TA) phonon modes. The unscreened TA-DP is found to dominate $S_{xx}^g$ over other mechanisms. The $S_{xx}^g$ is found to oscillate with the magnetic field where the lifting effect of the valley and spin degeneracies in MoS$_2$ monolayer has been clearly observed. An enhanced $S_{xx}^g$ with a peak value of $\sim1$~mV~K$^{-1}$ at about \mbox{$T=10$~K} is predicted, which is closer to the zero field experimental observation. In the Bloch-Gr\"{u}neisen regime the temperature dependence of $S_{xx}^g$ gives the power-law $S_{xx}^g\propto T^{\delta_e}$, where $\delta_e$ varies marginally around 3 and 5 for unscreened and screened couplings, respectively. In addition, $S_{xx}^g$ is smaller for larger electron density $n_e$. The power factor PF is found to increase with temperature $T$, decrease with $n_e$, and oscillate with $B$. The prediction of an increase of thermal conductivity with temperature and the magnetic field is responsible for the limit of the figure of merit ($ZT$). At a particular magnetic field and temperature, $ZT$ can be maximized by optimizing electron density. By fixing $n_e=10^{12}$~cm$^{-2}$, the highest $ZT$ is found to be $0.57$ at $T=5.8$~K and $B=12.1$~T. Our findings are compared with those in graphene and MoS$_2$ for the zero-magnetic field.

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