This paper studies a generic model for cooperative cognitive radio networks where the secondary user is equipped with a finite relay queue as well as a finite battery queue. Our prime objective is to characterize the stable throughput region. Nevertheless, the complete characterization of the stable throughput region for such system is notoriously difficult, since the computation of the steady state distribution of the two-dimensional Markov Chain (MC) model for both finite queues is prohibitively complex. We first propose an algorithm to characterize the stable throughput region numerically, and show its sheer computational complexity for large queue lengths. To lend tractability and explore the nature of design parameters optimization at the cognitive node, we next focus on two simpler systems, namely, finite battery queue with infinite relay queue and finite relay queue with infinite battery queue (referred henceforth as dominant system $1$ and $2$, respectively). For each proposed dominant system, we investigate the maximum service rate of the cognitive node subject to stability conditions. Despite the complexity of the formulated optimization problems, due to their non-convexity, we exploit the problems' structure to transform them into linear programs. Thus, we are able to solve them efficiently using standard linear programming solvers. Our numerical results demonstrate that, in practical systems, finite battery and relay queues achieve the same level of benefits of a system with infinite queue sizes, when their sizes are sufficiently large. They also reveal that the achievable stable throughput region significantly expands when the arrival rate of the energy harvesting process increases.