We investigate the properties of ∼ 7000 narrow-band selected galaxies with strong Hβ+[Oiii] and [Oii] nebular emission lines from the High-z Emission Line Survey (HiZELS) between z ∼ 0.8 − 5.0. Our sample covers a wide range in stellar mass (Mstellar ∼ 107.5−12.0 M), rest-frame equivalent widths (EWrest∼ 10 − 105 ˚A), and line luminosities (Lline ∼ 1040.5−43.2 erg s−1 ). We measure the Hβ+[Oiii]-selected stellar mass functions out to z ∼ 3.5 and find that both M? and φ? increases with cosmic time, which may be due to the [Oiii] selection including an increasing fraction of AGN at lower redshifts. The [Oii]-selected stellar mass functions show a constant M? ≈ 1011.6 M and a strong, increasing evolution with cosmic time in φ? in line with Hα studies. We also investigate the evolution of the EWrest as a function of redshift with a fixed mass range (109.5−10.0 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 z ∼ 3 for Hβ+[Oiii] and [Oii] emitters, respectively. This is the first time that the EWrest evolution has been directly measured for Hβ+[Oiii] and [Oii] emitters up to these redshifts. There is evidence for a slower evolution for z > 2 in the Hβ+[Oiii] EWrest and a decreasing trend for z > 3 in the [Oii] EWrest evolution, which would imply low [Oii] EW at the highest redshifts and higher [Oiii]/[Oii] 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 EWrest and [Oiii]/[Oii] evolution.