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The Space-Air-Ground Information Network (SAGIN) provides extensive coverage, enabling global connectivity across a diverse array of sensors, devices, and objects. These devices generate large amounts of data that require advanced analytics and decision making using artificial intelligence techniques. However, traditional deep learning approaches encounter drawbacks, primarily, the requirement to transmit substantial volumes of raw data to central servers, which raises concerns about user privacy breaches during transmission. Federated learning (FL) has emerged as a viable solution to these challenges, addressing both data volume and privacy issues effectively. Nonetheless, the deployment of FL faces its own set of obstacles, notably the excessive delay and energy consumption caused by the vast number of devices and fluctuating channel conditions. In this paper, by considering the heterogeneity of devices and the instability of the network state, the delay and energy consumption models of each round of federated training are established. Subsequently, we introduce a strategic node selection approach aimed at minimizing training costs. Building upon this, we propose an innovative, empirically driven Double Deep Q Network (DDQN)-based algorithm called low-cost node selection in federated learning (LCNSFL). The LCNSFL algorithm can assist edge servers in selecting the optimal set of devices to participate in federated training before the start of each round, based on the collected system state information. This paper culminates with a simulation-based comparison, showcasing the superior performance of LCNSFL against existing algorithms, thus underscoring its efficacy in practical applications.