To prevent elderly people from being socially isolated, encouraging their self-disclosure takes an important role. We discuss a use case of social robots in which they are deployed as mediators for humans that intermediate remote communication between elderly people and their family members or friends. The goal of this article is to present a design guideline for such social mediator robots based on results obtained from two studies in which a total of 741 elderly people participated. In study 1, we explored topics in dialogues and found that a social mediator robot could well encourage the self-disclosure of the elderly people, particularly in topics of which they usually feel resistance in talking to others (e.g., loss experiences). Thus, we confirmed the feasibility of the social mediator robot. Study 2 pursued the effective personality traits of the social mediator robot. We re-investigated a well-studied research question of matching robot personality to the user. The results provided more detailed knowledge as to similarity-attraction/repulsion than had been reported previously. Finally, design recommendations were discussed by considering the personality traits of the elderly users as well.