As the elderly population grows worldwide, living a healthy and full life as an older adult is becoming a topic of great interest. One key factor and severe challenge to maintaining quality of life in older adults is cognitive decline. Assistive robots for helping older adults have been proposed to solve issues such as social isolation and dependent living. Only a few studies have reported the positive effects of dialogue robots on cognitive function but conversation is being discussed as a promising intervention that includes various cognitive tasks. Existing dialogue robot-related studies have reported on placing dialogue robots in elderly homes and allowing them to interact with residents. However, it is difficult to reproduce these experiments since the participants’ characteristics influence experimental conditions, especially at home. Besides, most dialogue systems are not designed to set experimental conditions without on-site support. This study proposes a novel design method that uses a dialogue-based robot system for cognitive training at home. We define challenges and requirements to meet them to realize cognitive function training through daily communication. Those requirements are designed to satisfy detailed conditions such as duration of dialogue, frequency, and starting time without on-site support. Our system displays photos and gives original stories to provide contexts for dialogue that help the robot maintain a conversation for each story. Then the system schedules dialogue sessions along with the participant’s plan. The robot moderates the user to ask a question and then responds to the question by changing its facial expression. This question-answering procedure continued for a specific duration (4 min). To verify our design method’s effectiveness and implementation, we conducted three user studies by recruiting 35 elderly participants. We performed prototype-, laboratory-, and home-based experiments. Through these experiments, we evaluated current datasets, user experience, and feasibility for home use. We report on and discuss the older adults’ attitudes toward the robot and the number of turns during dialogues. We also classify the types of utterances and identify user needs. Herein, we outline the findings of this study, outlining the system’s essential characteristics to experiment toward daily cognitive training and explain further feature requests.