Graph Neural Networks (GNNs) show strong expressive power on graph data mining, by aggregating information from neighbors and using the integrated representation in the downstream tasks. The same aggregation methods and parameters for each node in a graph are used to enable the GNNs to utilize the homophily relational data. However, not all graphs are homophilic, even in the same graph, the distributions may vary significantly. Using the same convolution over all nodes may lead to the ignorance of various graph patterns. Furthermore, many existing GNNs integrate node features and structure identically, which ignores the distributions of nodes and further limits the expressive power of GNNs. To solve these problems, we propose Meta Weight Graph Neural Network (MWGNN) to adaptively construct graph convolution layers for different nodes. First, we model the Node Local Distribution (NLD) from node feature, topological structure and positional identity aspects with the Meta-Weight. Then, based on the Meta-Weight, we generate the adaptive graph convolutions to perform a node-specific weighted aggregation and boost the node representations. Finally, we design extensive experiments on real-world and synthetic benchmarks to evaluate the effectiveness of MWGNN. These experiments show the excellent expressive power of MWGNN in dealing with graph data with various distributions.