In this paper, we take advantage of the GAlaxy Evolution and Assembly (GAEA) semi-analytic model to analyse the origin of secondary dependencies in the local galaxy mass-gas metallicity relation. Our model reproduces quite well the trends observed in the local Universe as a function of galaxy star formation rate and different gas-mass phases. We show that the cold gas content (whose largest fraction is represented by the atomic gas phase) can be considered as the third parameter governing the scatter of the predicted mass-metallicity relation, in agreement with the most recent observational measurements. The trends can be explained with fluctuations of the gas accretion rates: a decrease of the gas supply leads to an increase of the gas metallicity due to star formation, while an increase of the available cold gas leads to a metallicity depletion. We demonstrate that the former process is responsible for offsets above the mass-metallicity relation, while the latter is responsible for deviations below the mass-metallicity relation. In low- and intermediate-mass galaxies, these negative offsets are primarily determined by late gas cooling dominated by material that has been previously ejected due to stellar feedback.