In this article, we propose an aspiration-based coevolution of link weights, and we explore how this affects the evolution of cooperation in the spatial prisoner's dilemma game. In particular, an individual will increase the weight of its link to its neighbors only if the payoff received via this interaction will exceed a pre-defined aspiration. Conversely, if the received payoff is below this aspiration, the link weight with the corresponding neighbor will be decreased. Our simulation results show that an appropriate aspiration level leads to a high-cooperation plateau, whereas a too high or a too low aspiration will impede the evolution of cooperation. We explain these findings with a comprehensive analysis of transition points and with a systematic analysis of typical configuration patterns. The presented results provide further theoretical insights with regards to the impact of different aspiration levels on cooperation in human societies.