AI assistants that use a voice user interface (VUI), such as AI speakers, have become popular in family homes. However, it is still unclear what roles the AI speaker can support within the family unit. We investigated the roles of an AI speaker as a family-shared technology. By conducting a one-week participatory user study, we discovered that family members' co-ownership toward the AI speaker was the key in the development of its family-oriented roles. Our findings showed seven domains of user expectations on these roles, and we realized that all the expectations can be represented as family cohesion. In addition, privacy awareness was emphasized regarding personal supports. Finally, we discuss a new perspective for AI speaker design and offer two suggestions: 1) leveraging human-likeness to develop its potential roles of supporting the unit of a family and 2) interpreting the home context to seamlessly connect family and personal supporting roles.