Code summaries are brief natural language descriptions of source code pieces. The main purpose of code summarization is to assist developers in understanding code and to reduce documentation workload. In this paper, we design a novel multi-task learning (MTL) approach for code summarization through mining the relationship between method code summaries and method names. More specifically, since a method’s name can be considered as a shorter version of its code summary, we first introduce the tasks of generation and informativeness prediction of method names as two auxiliary training objectives for code summarization. A novel two-pass deliberation mechanism is then incorporated into our MTL architecture to generate more consistent intermediate states fed into a summary decoder, especially when informative method names do not exist. To evaluate our deliberation MTL approach, we carried out a large-scale experiment on two existing datasets for Java and Python. The experiment results show that our technique can be easily applied to many state-of-the-art neural models for code summarization and improve their performance. Meanwhile, our approach shows significant superiority when generating summaries for methods with non-informative names.