  We report the discovery of COOL J1241+2219, a strongly-lensed galaxy at
redshift $z$=5.043$\pm$0.002 with observed magnitude $z_{AB}=20.47$, lensed by
a moderate-mass galaxy cluster at $z$=1.001$\pm$0.001. COOL J1241+2219 is the
brightest lensed galaxy currently known at optical and near-infrared
wavelengths at $z$ $\gtrsim$ 5; it is $\sim$5 times brighter than the prior
record-holder lensed galaxy, and several magnitudes brighter than the brightest
unlensed galaxies known at these redshifts. It was discovered as part of
COOL-LAMPS, a collaboration initiated to find strongly lensed systems in recent
public optical imaging data. We characterise the lensed galaxy, as well as the
central galaxy of the lensing cluster using ground-based $griz$JH imaging and
optical spectroscopy. We report model-based magnitudes, and derive stellar
masses, dust content, metallicity and star-formation rates via
stellar-population synthesis modeling. Our lens mass modeling, based on
ground-based imaging, implies a median source magnification of $\sim$30, which
puts the stellar mass and star formation rate (in the youngest age bin, closest
to the epoch of observation) at logM$_{*}$ = $10.11^{+0.21}_{-0.26}$ and SFR =
$27^{+13}_{-9}$ M$_{\odot}$/yr, respectively. We constrain a star formation
history for COOL J1241+2219 consistent with constant star formation across
$\sim$1 Gyr of cosmic time, and that places this galaxy on the high-mass end of
the star-forming main sequence. COOL J1241+2219 is 2-4 times more luminous than
a galaxy with the characteristic UV luminosity at these redshifts. The UV
continuum slope $\beta$= -2.2$\pm$0.2 places this galaxy on the blue side of
the observed distribution of galaxies at $z$=5, although the lack of Ly$\alpha$
emission indicates dust sufficient to suppress this emission.
