We study zero-temperature Glauber dynamics for Ising-like spin variable models in quenched random networks with random zero-magnetization initial conditions. In particular, we focus on the absorbing states of finite systems. While it has quite often been observed that Glauber dynamics lets the system be stuck into an absorbing state distinct from its ground state in the thermodynamic limit, very little is known about the likelihood of each absorbing state. In order to explore the variety of absorbing states, we investigate the probability distribution profile of the active link density after saturation as the system size N and (k) vary. As a result, we find that the distribution of absorbing states can be split into two self-averaging peaks whose positions are determined by (k), one slightly above the ground state and the other farther away. Moreover, we suggest that the latter peak accounts for a nonvanishing portion of samples when N goes to infinity while (k) stays fixed. Finally, we discuss the possible implications of our results on opinion dynamics models.