We present spectrophotometric data for a sample of 15 H I-selected dwarf galaxies in the Hydra I cluster (see Duc et al. 1999, Paper I). Long-slit spectra were obtained along an axis including the nucleus and the brightest H II region in each galaxy. The oxygen abundance was estimated from the emission lines using various methods. All dwarfs host one or several star-forming regions that are, for the most part, located outside the nucleus. We did not find in this survey any extremely metal-deficient objects, but we did observe some galaxies with a surprisingly high metallicity for their luminosity. In a following paper in this series, we will discuss the global properties of the H I-rich dwarfs in the Hydra I cluster and focus on their dependence on the environment. Based on observations collected at the European Southern Observatory, La Silla, Chile (ESO No 60.A-0791).