  We identify and phase a sample of 81 Cepheids in the maser-host galaxy NGC
4258 using the Large Binocular Telescope (LBT), and obtain calibrated mean
magnitudes in up to 4 filters for a subset of 43 Cepheids using archival HST
data. We employ 3 models to study the systematic effects of extinction, the
assumed extinction law, and metallicity on the Cepheid distance to NGC 4258. We
find a correction to the Cepheid colors consistent with a grayer extinction law
in NGC 4258 compared to the Milky Way ($R_V =4.9$), although we believe this is
indicative of other systematic effects. If we combine our Cepheid sample with
previously known Cepheids, we find a significant metallicity adjustment to the
distance modulus of $\gamma_1 = -0.61 \pm 0.21$ mag/dex, for the Zaritsky et
al. (1994) metallicity scale, as well as a weak trend of Cepheid colors with
metallicity. Conclusions about the absolute effect of metallicity on Cepheid
mean magnitudes appear to be limited by the available data on the metallicity
gradient in NGC 4258, but our Cepheid data require at least some metallicity
adjustment to make the Cepheid distance consistent with independent distances
to the LMC and NGC 4258. From our ensemble of models and the geometric maser
distance of NGC 4258 ($\mu_{N4258} = 29.40 \pm 0.06$ mag), we estimate
$\mu_{LMC} = 18.57 \pm 0.14$ mag ($51.82 \pm 3.23$ kpc).
