With the increase of spacecrafts, data relay satellite network (DRSN) plays an essential role in the space task transmission and processing through broad coverage and efficient information distribution of data relay satellites. However, the growing demand of space tasks and the limited number of satellite antenna lead to the conflicts between high-priority unpredictable tasks and common tasks with timely processing requirements, which further result in the decreasing scheduling efficiency of space tasks. In this paper, a dynamic scheduling scheme with task laxity (DSTL) is proposed to solve scheduling conflicts and enhance scheduling efficiency. Firstly, a task model in visible time window is analyzed according to the processing time limit of tasks and the intermittent connection nature of satellites under a constructed DRSN model. After that, the task laxity is defined by the deadline and execution time of tasks to judge the urgency of tasks, and the optimization problem is formulated to maximize the number of tasks and prioritize high priority tasks by calculating task laxity. Then, the DSTL scheme is designed via four stages, which are task preprocessing, task-resource matching, task-conflict assessment, task-resource updating. In DSTL, a task sorting method with task laxity is raised to arrange the priority of tasks; an urgent task scheduling algorithm (UTS) is put forward to prioritize unpredictable tasks; a task-conflict resolution algorithm (TCR) is brought up to assess conflicting tasks and reschedule them. Following that, an adaptive large neighborhood search algorithm combined with deadline-aware scheduling (ALNS-DAS) is presented to obtain the optimal scheduling scheme, which has the maximum number of completed tasks. The simulation results show that the DSTL can maximize the guarantee ratio of completed tasks and ensure timely processing with the increase of tasks demand.