The category {\bf FI} and its variants have been of great interest recently. Being a finitely generated {\bf FI}-module implies many desirable properties about sequences of symmetric group representations, in particular representation stability. We define a new combinatorial category analogous to {\bf FI} for the 0-Hecke algebra, denoted by $\mathcal{H}$, indexing sequences of representations of $H_n(0)$ as $n$ varies under suitable compatibility conditions. We then provide examples of $\mathcal{H}$-modules and use these to discuss some properties finitely generated $\mathcal{H}$-modules possess, including a new form of representation stability and eventually polynomial growth.