This paper formulates an optimality criterion for the selection of synchrophasor signals to be used in ambient mode estimators. This criterion, which is associated with each measured signal and each dominant mode, is defined as the asymptotic variance of the corresponding estimated mode damping ratio. The value of the criterion is computed directly from an estimated autoregressive moving average (ARMA) model. Because the online computation of the defined criterion (for each measured signal in the system) may be computationally expensive, a fast pre-selection method for initial signal ranking is formulated. The pre-selection method is used to effectively determine a set of the candidate signals for which the formal criterion is evaluated in a second stage. The methodology is illustrated using synthetic measurements from the KTH Nordic 32 and the IEEE 39-bus test systems.