This paper addresses the stabilization problem of rigid spacecraft attitude dynamics in the presence of external stochastic torques and an unknown timevarying delay in the measurement. By employing a linear delayed state feedback controller, a suitable Lyapunov-Krasovskii functional, and Ito's differential formula, a delay dependent stability condition in terms of a linear matrix inequality (LMI) is obtained whose solution gives the controller gain matrices. In addition, a quadratic cost function is applied to the derived LMI to achieve a suboptimal control performance for the system. An estimate of the region of attraction of the controlled system is also obtained, inside which the asymptotic stability of system is guaranteed in the mean-square sense. Finally, to show the effectiveness of the proposed controller, a set of simulations are performed on the attitude dynamics of a given spacecraft.