There is a nice preprint Universal thermopower of bad metals
Veljko Zlatic, G.R. Boyd, Jim Freericks
It contains calculations of the temperature and doping dependence of the thermoelectric power for the Falicov-Kimball model within the approximation of Dynamical-Mean Theory [DMFT].
This spinless fermion model is even "simpler" than the Hubbard model. Yet it captures some of the same physics, particularly the Mott metal-insulator transition. It also has the advantage that DMFT has an exact analytical solution. One does not need an "impurity solver", such as for the Hubbard model. There is an extensive Rev. Mod. Phys. on this, by Freericks and Zlatic.
Below I discuss one significant disadvantage of the model.
The figure below shows the calculated temperature dependence of the thermopower for several different dopings. The solid lines are the result from the Kubo formula [essentially exact] and the dashed line is the approximate Kelvin formula [the derivative of the chemical potential with respect to temperature].
Note that both the magnitude [of order k_B/e=80 microVolt/K] and non-monotonic temperature dependence are similar to what one sees in many strongly correlated electron materials. [Compare for example this post about heavy fermion compounds.]
Furthermore, it is striking that the Kelvin formula gives semi-quantitative results that are reliable.
However, when it comes to detailed comparison with experiment on actual materials, it is important to keep in mind a significant shortcoming of the Falicov-Kimball model. It does not seem to have a low-energy coherence scale associated with the formation of Fermi liquid quasi-particles. In many strongly correlated electron materials this energy scale is much less than the bare energy scale t, of the intersite hopping. In the Figure above one can see that the temperature dependence of the thermopower occurs on a scale of order some significant fraction of the hopping t. For example, in organic charge transfer salts this is of order 400 K, and in the cuprates t is of order 4000 K. In these materials the thermopower varies on a scale that is one order of magnitude smaller.
I thank Nandan Pakhira for bringing the preprint to my attention.