The extended Baryon Oscillation Spectroscopic Survey of the Sloan Digital Sky Survey (SDSS-IV/eBOSS) has an extensive quasar program that combines several selection methods. Among these, the photometric variability technique provides highly uniform samples, which are unaffected by the redshift bias of traditional optical-color selections, when z = 2.7-3.5 quasars cross the stellar locus or when host galaxy light affects quasar colors at z< 0.9. We present the variability selection of quasars in eBOSS, focusing on a specific program that led to a sample of 13 876 quasars to gdered = 22.5 over a 94.5 deg2 region in Stripe 82, which has an areal density 1.5 times higher than over the rest of the eBOSS footprint. We use these variability-selected data to provide a new measurement of the quasar luminosity function (QLF) in the redshift range of 0.68 2.2. Both models are constrained to be continuous at z = 2.2. They present a flattening of the bright-end slope at high redshift. The LEDE model indicates a reduction of the break density with increasing redshift, but the evolution of the break magnitude depends on the parameterization. The models are in excellent accord, predicting quasar counts that agree within 0.3% (resp., 1.1%) to g< 22.5 (resp., g< 23). The models are also in good agreement over the entire redshift range with models from previous studies.