Graph Neural Networks (GNNs) are vital for learning from graph-structured data, enabling applications in network analysis, recommendation systems, and speech analytics. Deploying them on edge devices like client PCs and laptops enhances real-time processing, privacy, and cloud independence. GNNs aid Retrieval-Augmented Generation (RAG) for Large Language Models (LLMs) and enable event-based vision tasks. However, irregular memory access, sparsity, and dynamic structures cause high latency and energy overhead on resource-constrained devices. While modern edge processors integrate CPUs, GPUs, and NPUs, NPUs designed for data-parallel tasks struggle with irregular GNN computations. We introduce GraNNite, the first hardware-aware framework optimizing GNN execution on commercial-off-the-shelf (COTS) SOTA DNN accelerators via a structured three-step methodology: (1) enabling NPU execution, (2) optimizing performance, and (3) trading accuracy for efficiency gains. Step 1 employs GraphSplit for workload distribution and StaGr for static aggregation, while GrAd and NodePad handle dynamic graphs. Step 2 boosts performance using EffOp for control-heavy tasks and GraSp for sparsity exploitation. Graph Convolution optimizations PreG, SymG, and CacheG reduce redundancy and memory transfers. Step 3 balances quality versus efficiency, where QuantGr applies INT8 quantization, and GrAx1, GrAx2, and GrAx3 accelerate attention, broadcast-add, and SAGE-max aggregation. On Intel Core Ultra AI PCs, GraNNite achieves 2.6X to 7.6X speedups over default NPU mappings and up to 8.6X energy gains over CPUs and GPUs, delivering 10.8X and 6.7X higher performance than CPUs and GPUs, respectively, across GNN models.