  The effects of an electron-phonon ($e$-ph) interaction on the thermoelectric
properties of Na$_x$CoO$_2$ are analyzed. By means of dynamical mean field
theory calculations we find that the $e$-ph coupling acts in a cooperative way
with the disorder, enhancing the effective binary disorder potential strength
on the Co sites, which stems from the presence or absence of a neighboring Na
atom. Hence, the inclusion of the $e$-ph coupling allows us to assume smaller
values of the binary disorder potential strength -- which for Na$_x$CoO$_2$ are
in fact also more reasonable. More generally, we can conclude that the
interplay between disorder effects and coupling to the lattice can be exploited
to engineer more efficient thermoelectric materials.
