With the development of telemedicine and edge computing, edge artificial intelligence (AI) will become a new development trend for smart medicine. On the other hand, nearly one-third of children suffer from sleep disorders. However, all existing sleep staging methods are for adults. Therefore, we adapted edge AI to develop a lightweight automatic sleep staging method for children using single-channel EEG. The trained sleep staging model will be deployed to edge smart devices so that the sleep staging can be implemented on edge devices which will greatly save network resources and improving the performance and privacy of sleep staging application. Then the results and hypnogram will be uploaded to the cloud server for further analysis by the physicians to get sleep disease diagnosis reports and treatment opinions. We utilized 1D convolutional neural networks (1D-CNN) and long short term memory (LSTM) to build our sleep staging model, named CSleepNet. We tested the model on our childrens sleep (CS) dataset and sleep-EDFX dataset. For the CS dataset, we experimented with F4-M1 channel EEG using four different loss functions, and the logcosh performed best with overall accuracy of 83.06% and F1-score of 76.50%. We used Fpz-Cz and Pz-Oz channel EEG to train our model in Sleep-EDFX dataset, and achieved an accuracy of 86.41% without manual feature extraction. The experimental results show that our method has great potential. It not only plays an important role in sleep-related research, but also can be widely used in the classification of other time sequences physiological signals.