  Direct observational searches for Population III (Pop III) stars at high
redshift are faced with the question of how to select the most promising
targets for spectroscopic follow-up. To help answer this, we use a large-scale
cosmological simulation, augmented with a new subgrid model that tracks the
fraction of pristine gas, to follow the evolution of high-redshift galaxies and
the Pop III stars they contain. We generate rest-frame ultraviolet (UV)
luminosity functions for our galaxies and find that they are consistent with
current $z \ge 7 $ observations. Throughout the redshift range $7 \le z \le 15$
we identify "Pop III-bright" galaxies as those with at least 75% of their flux
coming from Pop III stars. While less than 1% of galaxies brighter than $m_{\rm
UV, AB} = 31.4$ mag are Pop III--bright in the range $7\leq z \leq8$, roughly
17% of such galaxies are Pop III--bright at $z=9$, immediately before
reionization occurs in our simulation. Moving to $z=10$, $m_{\rm UV, AB} =
31.4$ mag corresponds to larger, more luminous galaxies and the Pop III-bright
fraction falls off to 5%. Finally, at the highest redshifts, a large fraction
(29% at $z=14$ and 41% at $z=15)$ of all galaxies are Pop III-bright regardless
of magnitude. While $m_{\rm UV, AB} = 31.4$ mag galaxies are extremely rare
during this epoch, we find that 13% of galaxies at $z = 14$ are Pop III-bright
with $m_{\rm UV, AB} \le 33$ mag, an intrisic magnitude within reach of the
James Webb Space Telescope using lensing. Thus, we predict that the best
redshift to search for luminous Pop III--bright galaxies is just before
reionization, while lensing surveys for fainter galaxies should push to the
highest redshifts possible.
