We investigate the properties of ∼7000 narrow-band selected galaxies with strong Hβ+[O III] and [O II] nebular emission lines from the High-z Emission-Line Survey between z ∼ 0.8 and 5.0. Our sample covers a wide range in stellar mass (M_(stellar) ∼ 10^(7.5–12.0) M_⊙), rest-frame equivalent widths (EW_(rest) ∼10–10^5 Å), and line luminosities (L_(line) ∼ 10^(40.5–43.2) erg s^(−1)). We measure the Hβ+[O III]-selected stellar mass functions out to z ∼ 3.5 and find that both M_⋆ and ϕ_⋆ increases with cosmic time. The [O II]-selected stellar mass functions show a constant M_⋆ ≈ 10^(11.6) M_⊙ and a strong, increasing evolution with cosmic time in ϕ_⋆ in line with Hα studies. We also investigate the evolution of the EW_(rest) as a function of redshift with a fixed mass range (10^(9.5–10.0) M_⊙) and find an increasing trend best represented by (1 + z)^(3.81 ± 0.14) and (1 + z)^(2.72 ± 0.19) up to z ∼ 2 and ∼3 for Hβ+[O III] and [O II] emitters, respectively. This is the first time that the EW_(rest) evolution has been directly measured for Hβ+[O III] and [O II] emitters up to these redshifts. There is evidence for a slower evolution for z > 2 in the Hβ+[O III] EW_(rest) and a decreasing trend for z > 3 in the [O II] EW_(rest) evolution, which would imply low [O II] EW at the highest redshifts and higher [O III]/[O II] line ratios. This suggests that the ionization parameter at higher redshift may be significantly higher than the local Universe. Our results set the stage for future near-IR space-based spectroscopic surveys to test our extrapolated predictions and also produce z > 5 measurements to constrain the high-z end of the EW_(rest) and [O III]/[O II] evolution.