Estimating the variance of the sample mean from a stochastic process is essential in assessing the quality of using the sample mean to estimate the population mean, which is the fundamental question in simulation experiments. Most existing studies for estimating the variance of the sample mean from simulation output assume that the simulation run length is known in advance. An interesting and open question is how to estimate the variance of the sample mean with limited memory space, reasonable computation time, and good statistical properties such as small mean-squared-error (mse), without knowing the simulation run length a priori. This paper proposes a finite-memory algorithm that satisfies the above good estimation criteria. Our findings show that the proposed algorithm improves over its competitors in terms of the mse criterion.