Graph Neural Architecture Search (GNAS) has achieved superior performance on various graph-structured tasks. However, existing GNAS studies overlook the applications of GNAS in resource-constraint scenarios. This paper proposes to design a joint graph data and architecture mechanism, which identifies important sub-architectures via the valuable graph data. To search for optimal lightweight Graph Neural Networks (GNNs), we propose a Lightweight Graph Neural Architecture Search with Graph SparsIfication and Network Pruning (GASSIP) method. In particular, GASSIP comprises an operation-pruned architecture search module to enable efficient lightweight GNN search. Meanwhile, we design a novel curriculum graph data sparsification module with an architecture-aware edge-removing difficulty measurement to help select optimal sub-architectures. With the aid of two differentiable masks, we iteratively optimize these two modules to efficiently search for the optimal lightweight architecture. Extensive experiments on five benchmarks demonstrate the effectiveness of GASSIP. Particularly, our method achieves on-par or even higher node classification performance with half or fewer model parameters of searched GNNs and a sparser graph.