Although link prediction on graphs has achieved great success with the development of graph neural networks (GNNs), the potential robustness under the edge noise is still less investigated. To close this gap, we first conduct an empirical study to disclose that edge noise naturally perturbs both input topology and target label, yielding severe performance degradation and representation collapse. To address this dilemma, we then propose an information-theory-guided principle, Robust Graph Information Bottleneck (RGIB), to extract reliable supervision signals and avoid representation collapse. Different from the general information bottleneck, RGIB further decouples and balances the mutual dependence among graph topology, target labels, and representation, building new learning objectives toward robust representation against edge noise. Two instantiations, RGIB-SSL and RGIB-REP, are explored to leverage the merits of different methodologies, i.e., self-supervised learning and data reparametrization, for implicit and explicit data denoising, respectively. Extensive experiments on six datasets and three GNNs with diverse noisy scenarios verify the effectiveness of our RGIB variants.