Technology-mediated communication tools are widely used to maintain connections between older adults and their remote family members. However, challenges often arise due to differences in their life rhythms and communication preferences, such as time zones, daily routines, or preferred platforms. To address these challenges, we propose using Ditto, a mimetic embodied agent, in video-call-like interactions between an older adult and a remote family member. When direct interaction is difficult, FamilyDitto can represent either party, providing a strong social presence and a personalized experience to the other person. To explore the potential of Ditto in supporting intergenerational communication, we conducted seven co-design workshops (n=27) with older adults and younger family members. Our thematic analysis reveals Ditto's potential roles as both a temporal bridge and an emotional proxy, identifies the tension between faithful and idealized representations, and emphasizes the importance of personalization to support unique family dynamics. We distill our findings into implications for designing mediated semi-synchronous communication, using idealized representation responsibly, and addressing the asymmetric motivations and comfort levels with Ditto across roles and ages. This study provides a foundation for mimetic AI technologies that enhance, rather than replace, human connection in remote intergenerational relationships.