  We present an analysis of the nuclear variability of $\sim28,000$ nearby
($z<0.15$) galaxies with Sloan Digital Sky Survey (SDSS) spectroscopy in Stripe
82. We construct light curves using difference imaging of SDSS g-band images,
which allows us to detect subtle variations in the central light output. We
select variable AGN by assessing whether detected variability is well-described
by a damped random walk model. We find 135 galaxies with AGN-like nuclear
variability. While most of the variability-selected AGNs have narrow emission
lines consistent with the presence of an AGN, a small fraction have narrow
emission lines dominated by star formation. The star-forming systems with
nuclear AGN-like variability tend to be low-mass
($M_{\ast}<10^{10}~M_{\odot}$), and may be AGNs missed by other selection
techniques due to star formation dilution or low-metallicities. We explore the
AGN fraction as a function of stellar mass, and find that the fraction of
variable AGN increases with stellar mass, even after taking into account the
fact that lower mass systems are fainter. There are several possible
explanations for an observed decline in the fraction of variable AGN with
decreasing stellar mass, including a drop in the supermassive black hole
occupation fraction, a decrease in the ratio of black hole mass to galaxy
stellar mass, or a change in the variability properties of lower-mass AGNs. We
demonstrate that optical photometric variability is a promising avenue for
detecting AGNs in low-mass, star formation-dominated galaxies, which has
implications for the upcoming Large Synoptic Survey Telescope.
