Data augmentation is necessary for graph representation learning due to the scarcity and noise present in graph data. Most of the existing augmentation methods overlook the context information inherited from the dataset as they rely solely on the graph structure for augmentation. Despite the success of some large language model-based (LLM) graph learning methods, they are mostly white-box which require access to the weights or latent features from the open-access LLMs, making them difficult to be democratized for everyone as existing LLMs are mostly closed-source for commercial considerations. To overcome these limitations, we propose a black-box context-driven graph data augmentation approach, with the guidance of LLMs -- DemoGraph. Leveraging the text prompt as context-related information, we task the LLM with generating knowledge graphs (KGs), which allow us to capture the structural interactions from the text outputs. We then design a dynamic merging schema to stochastically integrate the LLM-generated KGs into the original graph during training. To control the sparsity of the augmented graph, we further devise a granularity-aware prompting strategy and an instruction fine-tuning module, which seamlessly generates text prompts according to different granularity levels of the dataset. Extensive experiments on various graph learning tasks validate the effectiveness of our method over existing graph data augmentation methods. Notably, our approach excels in scenarios involving electronic health records (EHRs), which validates its maximal utilization of contextual knowledge, leading to enhanced predictive performance and interpretability.