The SDSS-IV/eBOSS has an extensive quasar program that combines several selection methods. Among these, the photometric variability technique provides highly uniform samples, 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 $z < 0.9$. Here, we present the variability selection of quasars in eBOSS, focusing on a specific program that led to a sample of 13,876 quasars to $g_{rm dered}=22.5$ over a 94.5 deg$^2$ region in Stripe 82, 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 $0.68<z<4.0$. Our sample is denser, reaches deeper than those used in previous studies of the QLF, and is among the largest ones. At the faint end, our QLF extends to $M_g(z!=!2)=-21.80$ at low redshift and to $M_g(z!=!2)=-26.20$ at $zsim 4$. We fit the QLF using two independent double-power-law models with ten free parameters each. The first model is a pure luminosity-function evolution (PLE) with bright-end and faint-end slopes allowed to be different on either side of $z=2.2$. The other is a simple PLE at $z<2.2$, combined with a model that comprises both luminosity and density evolution (LEDE) at $z>2.2$. Both models are constrained to be continuous at $z=2.2$. They present a flattening of the bright-end slope at large 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.
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