  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 $z\sim 4$. 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.
