© 2015. The American Astronomical Society. All rights reserved. As part of the Sloan Digital Sky Survey (SDSS) IV the extended Baryon Oscillation Spectroscopic Survey (eBOSS) will improve measurements of the cosmological distance scale by applying the Baryon Acoustic Oscillation (BAO) method to quasar samples. eBOSS will adopt two approaches to target quasars over 7500 deg2. First, a "CORE" quasar sample will combine the optical selection in ugriz using a likelihood-based routine called XDQSOz, with a mid-IR-optical color cut. eBOSS CORE selection (to g < 22 or r < 22) should return ∼70 deg-2 quasars at redshifts 0.9 < z < 2.2 and ∼7 deg-2z > 2.1 quasars. Second, a selection based on variability in multi-epoch imaging from the Palomar Transient Factory should recover an additional ∼3-4 deg-2z > 2.1 quasars to g < 22.5. A linear model of how imaging systematics affect target density recovers the angular distribution of eBOSS CORE quasars over 96.7% (76.7%) of the SDSS north (south) Galactic Cap area. The eBOSS CORE quasar sample should thus be sufficiently dense and homogeneous over 0.9 < z < 2.2 to yield the first few-percent-level BAO constraint near eBOSS quasars at z > 2.1 will be used to improve BAO measurements in the Lyα Forest. Beyond its key cosmological goals, eBOSS should be the next-generation quasar survey, comprising >500,000 new quasars and >500,000 uniformly selected spectroscopically confirmed 0.9 < z < 2.2 quasars. At the conclusion of eBOSS, the SDSS will have provided unique spectra for more than 800,000 quasars.