A stochastic thermodynamics of finite electrochemical reaction systems affected by external noise has been established in this article. Based on a general stochastic model of the irreversible electrode processes driven by parameter noise, both the fluctuation and the dissipation in this kind of important physico-chemical system are discussed. It shows that an additional polarization induced by external noise in irreversible electrode processes is inevitable. Furthermore, a perfect formalism to estimate quantitatively both the concentration polarization and the activation polarization induced by a Gaussian white noise in the steady irreversible electrode process has been proposed. It turns out that the order of this noise-driven polarization could arrive at the level as noise intensity even in thermodynamic limit. Some illustrations to calculate the polarizations driven by the additive noise and the multiplicative noise in different cases are also given categorically. The formalism as a stochastic measure of the polarization induced by Gaussian white noise proposed in this article is helpful to further study and utilize generally this electrochemical effect driven by noise in physical chemistry.