With the rise of 5G technology, Mobile (or Multi-Access) Edge Computing (MEC) has become crucial in modern network architecture. One key research area is the effective placement of edge nodes, which has attracted significant attention. Service providers strive to minimize deployment costs for these nodes within a network. Although many studies have explored optimal strategies for reducing these costs, most overlook the allocation of computational resources and the users’ tolerance for delays. These factors add complexity, making previous methods less adaptable. In this paper, we define the Cost Minimization in MEC Edge Node Placement problem. Our goal is to find the optimal strategy for deploying edge nodes that minimize costs while cater to users’ delay tolerance limits. We prove the NP-hardness of this problem and provide a range of solutions, including Cluster-based Mixed Integer Programming, Coverage First Search, and Distance-Aware Coverage First Search, to address this challenge effectively and efficiently. Additionally, we propose a fine-grained optimization approach for allocating computational resources to edge nodes based on user service requests, significantly lowering deployment costs. Extensive experiments on a large-scale real-world dataset show that our solutions outperform the state-of-the-art in efficiency, effectiveness, and scalability.