  We characterize the luminosity distribution, halo mass dependence, and
redshift evolution of red galaxies in galaxy clusters using the SDSS Data
Release 8 RedMaPPer cluster sample. We propose a simple prescription for the
relationship between the luminosity of both central and satellite galaxies and
the mass of their host halos, and show that this model is well-fit by the data.
Using a larger galaxy cluster sample than previously employed in the
literature, we find that the luminosities of central galaxies scale as $\langle
\log L \rangle \propto A_L \log (M_{200b})$, with $A_L=0.39\pm0.04$, and that
the scatter of the central--galaxy luminosity at fixed $M_{200b}$ (
$\sigma_{\log L|M}$) is $0.23 ^{+0.05}_{-0.04}$ dex, with the error bar
including systematics due to miscentering of the cluster finder, photometry,
and photometric redshift estimation. Our data prefers a positive correlation
between the luminosity of central galaxies and the observed richness of
clusters at a fixed halo mass, with an effective correlation coefficient
$d_{\rm{eff}}=0.36^{+0.17}_{-0.16}$. The characteristic luminosity of
satellites becomes dimmer from $z=0.3$ to $z=0.1$ by $\sim 20\%$ after
accounting for passive evolution. We estimate the fraction of galaxy clusters
where the brightest galaxy is not the central to be $P_{\rm{BNC}} \sim 20\%$.
We discuss implications of these findings in the context of galaxy evolution
and the galaxy--halo connection.
