  NGC 4258 is one of the most important anchors for calibrating the Cepheid
period--luminosity relations (PLRs) owing to its accurate distance measured
from water maser motions. We expand on previous efforts and carry out a new
Cepheid search in this system using the Hubble Space Telescope (HST). We
discover and measure a sample of 669 Cepheids in four new and two archival NGC
4258 fields, doubling the number of known Cepheids in this galaxy and obtaining
an absolute calibration of their optical PLRs. We determine a Wesenheit PLR of
$-2.574(\pm0.034) -3.294(\pm0.042) \log P$, consistent with an independent
Large Magellanic Cloud (LMC) calibration at the level of $0.032\pm0.044$~mag in
its zeropoint, after accounting for a metallicity dependence of
$-0.20\pm0.05$~mag\,dex$^{-1}$ (Riess et al. 2006). Our determination of the
PLR slope also agrees with the LMC-based value within their uncertainties. We
attempt to characterize the metallicity effect of Cepheid PLRs using only the
NGC 4258 sample, but a relatively narrow span of abundances limits our
sensitivity and yields a Wesenheit zero-point dependence of $-0.07 \pm 0.21$
mag\,dex$^{-1}$. The Cepheid measurements presented in this study have been
used as part of the data to derive the Hubble constant in a companion paper by
the SH0ES team.
