Motivated by the fact that system parameter mismatch occurs in real-world sensing environments, this paper addresses power allocation for robust distributed Best-Linear-Unbiased-Estimation (BLUE) that takes account of the uncertainty in the local sensing noise variance. We adopt the Bayesian philosophy, wherein the sensing noise variance follows a statistical distribution widely used in the literature, and the communication channels between sensor nodes and the fusion center (FC) are assumed to be i.i.d. Rayleigh fading. To facilitate analysis, we propose to use the average reciprocal mean square error (ARMSE), averaged with respect to the distributions of sensing noise variance and fading channels, as the distortion metric. A fundamental inequality characterizing the relationship between ARMSE and the average mean square error (AMSE) is established. While the exact formula for ARMSE is difficult to find, we derive an associated closed-form lower bound which involves the complicated incomplete gamma function. To further ease analysis, we further derive a key inequality that specifies the range of the ARMSE lower bound. Particularly, it is shown that the boundary points of this inequality are characterized by a common quantity, which involves the Gaussian-tail function and is thus more analytically appealing. By conducting maximization of such a function, suboptimal sensor allocation factors are analytically derived. Computer simulation is used to evidence the effectiveness of the proposed robust power allocation scheme.