Turbulence in stable boundary layers is typically unsteady and intermittent. The study implements a stochastic modelling approach to represent unsteady mixing possibly associated with intermittency of turbulence and with unresolved fluid motions such as dirty waves or drainage flows. The stochastic parameterization is introduced by randomizing the mixing lengthscale used in a Reynolds average Navier-Stokes (RANS) model with turbulent kinetic energy closure, resulting in a stochastic unsteady RANS model. The randomization alters the turbulent momentum diffusion and accounts for sporadic events of possibly unknown origin that cause unsteady mixing. The paper shows how the proposed stochastic parameterization can be integrated into a RANS model used in weather-forecasting and its impact is analyzed using neutrally and stably stratified idealized numerical case studies. The simulations show that the framework can successfully model intermittent mixing in stably stratified conditions, and does not alter the representation of neutrally stratified conditions. It could thus present a way forward for dealing with the complexities of unsteady flows in numerical weather prediction or climate models.