Recently, Nicholls et al. (2012ApJ...752..148N), inspired by in situ observations of solar system astrophysical plasmas, suggested that the electrons in H II regions are characterized by a {kappa}-distribution of energies rather than a simple Maxwell-Boltzmann distribution. Here, we have collected together new atomic data within a modified photoionization code to explore the effects of both the new atomic data and the {kappa}-distribution on the strong-line techniques used to determine chemical abundances in H II regions. By comparing the recombination temperatures (T_rec_) with the forbidden line temperatures (T_FL_), we conclude that {kappa}~20. While representing only a mild deviation from equilibrium, this result is sufficient to strongly influence abundances determined using methods that depend on measurements of the electron temperature from forbidden lines. We present a number of new emission line ratio diagnostics that cleanly separate the two parameters determining the optical spectrum of H II regions--the ionization parameter q or {} and the chemical abundance, 12+log(O/H). An automated code to extract these parameters is presented. Using the homogeneous data set from van Zee et al. 1998, Cat. J/AJ/116/2805, we find self-consistent results between all of these different diagnostics. The systematic errors between different line ratio diagnostics are much smaller than those found in the earlier strong-line work. Overall, the effect of the {kappa}-distribution on the strong-line abundances derived solely on the basis of theoretical models is rather small.