  Properties of galaxies vary systematically with the mass of their parent dark
matter halos. This basic galaxy - halo connection shows a fair amount of
scatter whose origin is not fully understood. Here, we study how differences in
the halo assembly history affect central galaxies in low mass (M_halo < 10^12
M_sun) halos at z=2-6 with the help of the MassiveFIRE suite of cosmological
simulations. In contrast to previous works that tie galaxy properties to halo
concentration and halo formation redshift, we focus on halo growth rate as a
measure of assembly history. We find that, at fixed halo mass, faster growing
halos tend to have lower stellar masses and higher SFRs per unit stellar mass
but similar overall SFRs. We provide a simple explanation for these findings
with the help of an analytic model that captures approximately the behavior of
our hydrodynamical simulations. Specifically, among halos of a given current
mass, quickly growing halos have lower stellar masses (and thus higher sSFRs)
because they were less massive and had comparably lower cold gas masses and
SFRs in the past than slowly growing halos. By combining these findings with
estimates for the scatter of the halo growth rate, we show that variations in
growth rate at fixed halo mass may largely explain the scatter of the stellar
mass - halo mass relation. In contrast, halo growth variations likely play only
a minor role in the scatter of the star forming sequence in low mass galaxies.
