Graph Neural Networks (GNNs) have achieved state-of-the-art performance in various graph-based learning tasks. However, enabling privacy-preserving GNNs in encrypted domains, such as under Fully Homomorphic Encryption (FHE), typically incurs substantial computational overhead, rendering real-time and privacy-preserving inference impractical. In this work, we propose DESIGN (EncrypteD GNN Inference via sErver-Side Input Graph pruNing), a novel framework for efficient encrypted GNN inference. DESIGN tackles the critical efficiency limitations of existing FHE GNN approaches, which often overlook input data redundancy and apply uniform computational strategies. Our framework achieves significant performance gains through a hierarchical optimization strategy executed entirely on the server: first, FHE-compatible node importance scores (based on encrypted degree statistics) are computed from the encrypted graph. These scores then guide a homomorphic partitioning process, generating multi-level importance masks directly under FHE. This dynamically generated mask facilitates both input graph pruning (by logically removing unimportant elements) and a novel adaptive polynomial activation scheme, where activation complexity is tailored to node importance levels. Empirical evaluations demonstrate that DESIGN substantially accelerates FHE GNN inference compared to state-of-the-art methods while maintaining competitive model accuracy, presenting a robust solution for secure graph analytics. Our implementation is publicly available at https://github.com/LabRAI/DESIGN.