The 4 Ms Chandra Deep Field-South (CDF-S) and other deep X-ray surveys have been highly effective at selecting active galactic nuclei (AGN). However, cosmologically distant low-luminosity AGN (LLAGN) have remained a challenge to identify due to significant contribution from the host galaxy. We identify long-term X-ray variability (~month-years, observed frame) in 20 of 92 CDF-S galaxies spanning redshifts z~0.08-1.02 that do not meet other AGN selection criteria. We show that the observed variability cannot be explained by X-ray binary populations or ultraluminous X-ray sources, so the variability is most likely caused by accretion onto a supermassive black hole. The variable galaxies are not heavily obscured in general, with a stacked effective power-law photon index of Gamma_stack~1.93+/-0.13, and are therefore likely LLAGN. The LLAGN tend to lie a factor of ~6-80 below the extrapolated linear variability-luminosity relation measured for luminous AGN. This may be explained by their lower accretion rates. Variability-independent black-hole mass and accretion-rate estimates for variable galaxies show that they sample a significantly different black-hole mass-accretion rate space, with masses a factor of 2.4 lower and accretion rates a factor of 22.5 lower than variable luminous AGN at the same redshift. We find that an empirical model based on a universal broken power-law PSD function, where the break frequency depends on SMBH mass and accretion rate, roughly reproduces the shape, but not the normalization, of the variability-luminosity trends measured for variable galaxies and more luminous AGN.
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