We consider the critical properties of the two-point resistance and its fluctuations due to microscopic noise in a randomly diluted resistor network near the percolation threshold pc. We introduce a n×m replicated Hamiltonian in order to treat separately the configuration average over the randomly occupied bonds denoted [ ]av and the average over probability distribution function of the fluctuating microscopic bond conductance, denoted { }f. We evaluate a family of exponents {ψl} (l=2,3,. . .) whose values are 1+O(ε) with ε=6-d where d is the spatial dimensionality. Each ψl governs the critical behavior of the lth cumulant of the resistance between the sites x,x’ conditionally averaged subject to the sites being in the same cluster such that C¯R (l)(x,x’)~‖x-x'‖ψl/vp for p near pc, where νp is the correlation-length exponent for percolation. Furthermore, ψ2=1+ε/105 determines the dependence of the variance of the resistance in a finite network on size L as Lψ2/vp.