This paper investigates an inverse random source problem for stochastic evolution equations, including stochastic heat and wave equations, with the unknown source modeled as $g(x)f(t)\dot{W}(t)$. The research commences with the establishment of the well-posedness of the corresponding stochastic direct problem. Under suitable regularity conditions, the existence of stochastic strong solutions for both the stochastic heat and wave equations is demonstrated. For the inverse problem, the objective is to uniquely recover the strength $|f(t)|$ of the time-dependent component of the source from the boundary flux on a nonempty open subset. The uniqueness of the recovery for both the stochastic heat and wave equations is proven, and several numerical examples are given to verify the theoretical results.