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Resting state brain functional connectivity quantifies the similarity between brain regions, each of which consists of voxels at which dynamic signals are acquired via neuroimaging techniques such as blood-oxygen-level-dependent signals in functional magnetic resonance imaging. Pearson correlation and similar metrics have been adopted by neuroscientists to estimate inter-regional connectivity, usually after averaging of signals within regions. However, dependencies between signals within each region and the presence of noise could contaminate such inter-regional correlation estimates. We propose a mixed-effects model with a novel covariance structure that explicitly isolates the different sources of variability in the observed BOLD signals, including correlated regional signals, local spatiotemporal variability, and measurement error. Methods for tackling the computational challenges associated with restricted maximum likelihood estimation will be discussed. Large sample properties are discussed and used for uncertainty quantification. Simulation results demonstrate that the parameters of the proposed model parameters can be accurately estimated and is superior to the Pearson correlation of averages in the presence of spatiotemporal noise. The proposed model is also applied to a real data set of BOLD signals collected from rats to construct individual brain networks.