We present a study of galaxies showing mid-infrared variability in data taken in the deepest Spitzer/MIPS 24{mu}m surveys in the Great Observatory Origins Deep Survey South field. We divide the data set in epochs and subepochs to study the long-term (months-years) and the short-term (days) variability. We use a {chi}^2^-statistics method to select active galactic nucleus (AGN) candidates with a probability <=1% that the observed variability is due to statistical errors alone. We find 39 (1.7% of the parent sample) sources that show long-term variability and 55 (2.2% of the parent sample) showing short-term variability. That is, 0.03sources/arcmin^2^ for both, long-term and short-term variable sources. After removing the expected number of false positives inherent to the method, the estimated percentages are 1.0 and 1.4% of the parent sample for the long term and short term, respectively. We compare our candidates with AGN selected in the X-ray and radio bands, and AGN candidates selected by their IR emission. Approximately, 50% of the MIPS (Multiband Imaging Photometer for Spitzer) 24{mu}m variable sources would be identified as AGN with these other methods. Therefore, MIPS 24{mu}m variability is a new method to identify AGN candidates, possibly dust obscured and low-luminosity AGN, that might be missed by other methods. However, the contribution of the MIPS 24{mu}m variable identified AGN to the general AGN population is small (<=13%) in GOODS-South.