The global aging population has led to an increased prevalence of mild cognitive impairment (MCI), a condition that often progresses to dementia. Dementia, an irreversible disease, is characterized by progressive deterioration in cognitive domains such as complex attention, executive function, learning and memory, language, perceptual-motor skills, and social cognition. The DSM-5 includes social cognition as a new category under neurocognitive disorders (NCD), highlighting the significant impact of social interaction on cognitive health and the increased risk of dementia associated with social isolation. As the demand for care increases, human-robot interaction (HRI) is expected to become more prevalent in various settings, including hospitals, communities, and homes. This systematic review examines fifteen studies published over the past decade to identify strategies for improving social interaction interventions for elderly individuals with MCI using HRI. The review provides an overview of the human-robot social interaction experience and explores methods to enhance (1) robot usability through individual and design factors, (2) intention to use robots by addressing acceptability and utility, and (3) level of engagement by considering interest, interaction tools, and preparation for interacting with robots. The review concludes by discussing the barriers that affect HRI and providing recommendations for future research to optimize the potential of HRI in supporting the cognitive and social well-being of elderly individuals with MCI.