Despite wide recognition of the value, expertise, and support that patient-peers provide in a variety of health contexts, mechanisms to design and enable peer support in the inpatient setting have not been sufficiently explored. To better understand the opportunities for an inpatient peer support tool, we surveyed 100 adult patients and caregivers, and conducted follow-up, semi-structured interviews with 15 adult patients. In this paper, we describe five key peer support needs that our adult patient participants expressed: (1) adjusting to the hospital environment, (2) understanding and normalizing medical care, (3) communicating with providers, (4) reporting and preventing medical errors, and (5) empowering peers. In addition, we identify privacy concerns, situational impairments, and communication mode as barriers to, and preferences for, interacting with peers. Based on our findings, we discuss intelligent peer matching and aggregating peer data as design recommendations for future inpatient peer support tools.