  We consider systems of stochastic evolutionary equations of the $p$-Laplace
type. We establish convergence rates for a finite-element based space-time
approximation, where the error is measured in a suitable quasi-norm. Under
natural regularity assumptions on the solution, our main result provides linear
convergence in space and convergence of order $\alpha$ in time for all
$\alpha\in(0,\frac{1}{2})$. The key ingredient of our analysis is a random
time-grid, which allows us to compensate for the lack of time regularity. Our
theoretical results are confirmed by numerical experiments.
