As astronomers increasingly exploit the information available in the time domain, spectroscopic variability in particular opens broad new channels of investigation. Here we describe the selection algorithms for all targets intended for repeat spectroscopy in the Time Domain Spectroscopic Survey (TDSS), part of the extended Baryon Oscillation Spectroscopic Survey within the Sloan Digital Sky Survey (SDSS)-IV. Also discussed are the scientific rationale and technical constraints leading to these target selections. The TDSS includes a large "repeat quasar spectroscopy" (RQS) program delivering similar to 13,000 repeat spectra of confirmed SDSS quasars, and several smaller "few-epoch spectroscopy" (FES) programs targeting specific classes of quasars as well as stars. The RQS program aims to provide a large and diverse quasar data set for studying variations in quasar spectra on timescales of years, a comparison sample for the FES quasar programs, and an opportunity for discovering rare, serendipitous events. The FES programs cover a wide variety of phenomena in both quasars and stars. Quasar FES programs target broad absorption line quasars, high signal-to-noise ratio normal broad line quasars, quasars with double-peaked or very asymmetric broad emission line profiles, binary supermassive black hole candidates, and the most photometrically variable quasars. Strongly variable stars are also targeted for repeat spectroscopy, encompassing many types of eclipsing binary systems, and classical pulsators like RR Lyrae. Other stellar FES programs allow spectroscopic variability studies of active ultracool dwarf stars, dwarf carbon stars, and white dwarf/M dwarf spectroscopic binaries. We present example TDSS spectra and describe anticipated sample sizes and results.