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Graph data is ubiquitous in academia and industry, from social networks to bioinformatics. The pervasiveness of graphs today has raised the demand for algorithms that can answer various questions: Which products would a user like to purchase given her order list? Which users are buying fake followers to increase their public reputation? Myriads of new graph mining algorithms are proposed every year to answer such questions - each with a distinct problem formulation, computational time, and memory footprint. This lack of unity makes it difficult for a practitioner to compare different algorithms and pick the most suitable one for a specific application. These challenges - even more severe for non-experts - create a gap in which state-of-the-art techniques developed in academic settings fail to be optimally deployed in real-world applications. To bridge this gap, we propose AUTOGM, an automated system for graph mining algorithm development. We first define a unified framework UNIFIEDGM that integrates various message-passing based graph algorithms, ranging from conventional algorithms like PageRank to graph neural networks. Then UNIFIEDGM defines a search space in which five parameters are required to determine a graph algorithm. Under this search space, AUTOGM explicitly optimizes for the optimal parameter set of UNIFIEDGM using Bayesian Optimization. AUTOGM defines a novel budget-aware objective function for the optimization to incorporate a practical issue - finding the best speed-accuracy trade-off under a computation budget - into the graph algorithm generation problem. Experiments on real-world benchmark datasets demonstrate that AUTOGM generates novel graph mining algorithms with the best speed/accuracy trade-off compared to existing models with heuristic parameters.