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Ride-hailing demand prediction is an essential task in spatial-temporal data mining. Accurate Ride-hailing demand prediction can help to pre-allocate resources, improve vehicle utilization and user experiences. Graph Convolutional Networks (GCN) is commonly used to model the complicated irregular non-Euclidean spatial correlations. However, existing GCN-based ride-hailing demand prediction methods only assign the same importance to different neighbor regions, and maintain a fixed graph structure with static spatial relationships throughout the timeline when extracting the irregular non-Euclidean spatial correlations. In this paper, we propose the Spatial-Temporal Dynamic Graph Attention Network (STDGAT), a novel ride-hailing demand prediction method. Based on the attention mechanism of GAT, STDGAT extracts different pair-wise correlations to achieve the adaptive importance allocation for different neighbor regions. Moreover, in STDGAT, we design a novel time-specific commuting-based graph attention mode to construct a dynamic graph structure for capturing the dynamic time-specific spatial relationships throughout the timeline. Extensive experiments are conducted on a real-world ride-hailing demand dataset, and the experimental results demonstrate the significant improvement of our method on three evaluation metrics RMSE, MAPE and MAE over state-of-the-art baselines.