  Minimal free resolutions of graded modules over a noetherian polynomial ring
have been attractive objects of interest for more than a hundred years. We
introduce and study two natural extensions in the setting of graded modules
over a polynomial OI-algebra, namely minimal and width-wise minimal free
resolutions. A minimal free resolution of an OI-module can be characterized by
the fact that the free module in every fixed homological degree, say $i$, has
minimal rank among all free resolutions of the module. We show that any
finitely generated graded module over a noetherian polynomial OI-algebra admits
a graded minimal free resolution, and that it is unique. A width-wise minimal
free resolution is a free resolution that provides a minimal free resolution of
a module in every width. Such a resolution is necessarily minimal. Its
existence is not guaranteed. However, we show that certain monomial OI-ideals
do admit width-wise minimal free or, more generally, width-wise minimal flat
resolutions. These ideals include families of well-known monomial ideals such
as Ferrers ideals and squarefree strongly stable ideals. The arguments rely on
the theory of cellular resolutions.
