  Gy\'arf\'as famously showed that in every $r$-coloring of the edges of the
complete graph $K_n$, there is a monochromatic connected component with at
least $\frac{n}{r-1}$ vertices. A recent line of study by Conlon, Tyomkyn, and
the second author addresses the analogous question about monochromatic
connected components with many edges. In this paper, we study a generalization
of these questions for $k$-uniform hypergraphs. Over a wide range of extensions
of the definition of connectivity to higher uniformities, we provide both upper
and lower bounds for the size of the largest monochromatic component that are
tight up to a factor of $1+o(1)$ as the number of colors grows. We further
generalize these questions to ask about counts of vertex $s$-sets contained
within the edges of large monochromatic components. We conclude with more
precise results in the particular case of two colors.
