In real-world recommendation scenarios, users typically engage with platforms through multiple types of behavioral interactions. Multi-behavior recommendation algorithms aim to leverage various auxiliary user behaviors to enhance prediction for target behaviors of primary interest (e.g., buy), thereby overcoming performance limitations caused by data sparsity in target behavior records. Current state-of-the-art approaches typically employ hierarchical design following either cascading (e.g., view$\rightarrow$cart$\rightarrow$buy) or parallel (unified$\rightarrow$behavior$\rightarrow$specific components) paradigms, to capture behavioral relationships. However, these methods still face two critical challenges: (1) severe distribution disparities across behaviors, and (2) negative transfer effects caused by noise in auxiliary behaviors. In this paper, we propose a novel model-agnostic Hierarchical Graph Information Bottleneck (HGIB) framework for multi-behavior recommendation to effectively address these challenges. Following information bottleneck principles, our framework optimizes the learning of compact yet sufficient representations that preserve essential information for target behavior prediction while eliminating task-irrelevant redundancies. To further mitigate interaction noise, we introduce a Graph Refinement Encoder (GRE) that dynamically prunes redundant edges through learnable edge dropout mechanisms. We conduct comprehensive experiments on three real-world public datasets, which demonstrate the superior effectiveness of our framework. Beyond these widely used datasets in the academic community, we further expand our evaluation on several real industrial scenarios and conduct an online A/B testing, showing again a significant improvement in multi-behavior recommendations. The source code of our proposed HGIB is available at https://github.com/zhy99426/HGIB.