Software architectures are usually meticulously designed to address multiple quality concerns and support long-term maintenance. However, due to the imbalance between the cost and value for developers to document design rationales (i.e., the design alternatives and the underlying arguments for making or rejecting decisions), these rationales are often obsolete or even missing. The lack of design knowledge has motivated a number of studies to extract design information from various platforms in recent years. Unfortunately, despite the wealth of discussion records related to design information provided by platforms like open-source communities, existing research often overlooks the underlying arguments behind alternatives due to challenges such as the intricate semantics of discussions and the lack of benchmarks for design rationale extraction. In this paper, we propose a novel method, named by DRMiner, to automatically mine latent design rationales from developers' live discussion in open-source community (i.e., issue logs in Jira). To better identify solutions and the arguments supporting them, DRMiner skillfully decomposes the problem into multiple text classification tasks and tackles them using prompt tuning of language models and customized text-related features. To evaluate DRMiner, we acquire issue logs from Cassandra, Flink, and Solr repositories in Jira, and then annotate and process them under a rigorous scheme, ultimately forming a dataset for design rationale mining. Experimental results show that DRMiner achieves an F1 score of 65% for mining design rationales, outperforming all baselines with a 7% improvement over GPT-4.0. Furthermore, we investigate the usefulness of the design rationales mined by DRMiner for automated program repair (APR) and find that the design rationales significantly enhance APR, achieving 14 times higher full-match repairs on average.