He II λ4686 is identified in the spectra of nine of the original “Green Peas,” a type of compact star-forming galaxy characterized by low mass; low metallicity; strong [O III] λλ4959, 5007; and redshifts in the range of ̃0.1-0.4. Measured λ4686/Hβ ratios are roughly 1-2%, consistent with photoionization by Wolf-Rayet stars. Emission-line intensities are measured from Sloan Digital Sky Survey spectra for 71 Green Peas and are used to determine Te-based abundances of O, N, Ne, S, and He. Neon abundances confirm the mass-metallicity relation inferred from O/H. The N/O ratio is roughly constant with O/H, and the average N/O is evidence of a modest nitrogen enhancement compared with other low-metallicity galaxies. Nitrogen enrichment could be due to Wolf-Rayet stars or to intermediate-mass stars during a previous quiescent period. The Te-based abundances allow a reevaluation of some of the strong-line methods favored for estimating O/H or N/O in large spectroscopic surveys of star-forming galaxies. Photoionization by Wolf-Rayet stars raises questions about the validity of strong-line methods based on [N II]/Hα, [N II]/[O III], or [N II]/[S II], as those line ratios are known to be ionization-sensitive. Analysis of these measurements shows that ionization, low metallicity, and the small variation in important line ratios in the Green Pea spectra all affect the behavior of one or more of the N2, O3N2, N2O2 and N2S2 strong-line methods. The previous findings for trends in O/H and N/O in the Green Peas can be reproduced and the discrepancies can be explained. In particular, the reported increase of N/O with O/H appears to be a bias introduced by combining N2 with N2S2. N2O2 does not give valid results in the Green Peas, while N2 and N2S2 do, although the calibrations of the N2 and N2S2 methods based on Green Pea abundances are different from the existing calibrations based primarily on abundances in extragalactic H II regions and H II galaxies.