We present precision radial velocity (RV) data that reveal a Super-Earth mass planet and two probable additional planets orbiting the bright nearby G0V star HD 1461. Our 12.8 years of Keck High Resolution Echelle Spectrometer precision RVs indicate the presence of a 7.4 M _⊕ planet on a 5.77 day orbit. The data also suggest, but cannot yet confirm, the presence of outer planets on low-eccentricity orbits with periods of 446.1 and 5017 days, and projected masses (Msin i) of 27.9 and 87.1 M _⊕, respectively. Test integrations of systems consistent with the RV data suggest that the configuration is dynamically stable. We present a 12.2 year time series of photometric observations of HD 1461, which comprise 799 individual measurements, and indicate that it has excellent long-term photometric stability. However, there are small amplitude variations with periods comparable to those of the suspected second and third signals in the RVs near 5000 and 446 days, thus casting some suspicion on those periodicities as Keplerian signals. If the 5.77 day companion has a Neptune-like composition, then its expected transit depth is of order d~0.5 mmag. The geometric a priori probability of transits is ~8%. Phase folding of the ground-based photometry shows no indication that transits of the 5.77 day companion are occurring, but high-precision follow-up of HD 1461 during upcoming transit phase windows will be required to definitively rule out or confirm transits. This new system joins a growing list of solar-type stars in the immediate galactic neighborhood that are accompanied by at least one Neptune (or lower) mass planets having orbital periods of 50 days or less.