The Metaverse is redefining digital interactions by merging physical, virtual, and social dimensions, yet its effects on social networking remain largely unexplored. This work examines the role of independent avatars (autonomous digital entities capable of managing social interactions on behalf of users), to optimize social time allocation and reshape Metaverse-based Online Social Networks. We propose a novel computational model that integrates a quantitative and realistic representation of user social life, grounded in evolutionary anthropology, with a framework for avatar-mediated interactions. Our model quantifies the effectiveness of a partial replacement of in-person interactions with independent avatar interactions. Additionally, it accounts for social conflicts and specific socialization constraints. We leverage our model to explore the benefits and trade-offs of an avatar-augmented social life in the Metaverse. Since the exact problem formulation leads to an NP-hard optimization problem when incorporating avatars into the social network, we tackle this challenge by introducing a heuristic solution. Through simulations, we compare avatar-mediated and non-avatar-mediated social networking, demonstrating the potential of independent avatars to enhance social connectivity and efficiency. Our findings provide a foundation for optimizing Metaverse-based social interactions, as well as useful insights for future digital social network design.