Social media data (SMD) offer researchers new opportunities to leverage those data for their work in broad areas such as public opinion, digital culture, labor trends, and public health. The success of efforts to save SMD for reuse by researchers will depend on aligning data management and archiving practices with evolving norms around the capture, use, sharing, and security of datasets. This paper presents an initial foray into understanding how established practices for managing and preserving data should adapt to demands from researchers who use and reuse SMD, and from people who are subjects in SMD. We examine the data management practices of researchers who use SMD through a survey, and we analyze published articles that used data from Twitter. We discuss how researchers describe their data management practices and how these practices may differ from the management of conventional data types. We explore conceptual, technical, and ethical challenges for data archives based on the similarities and differences between SMD and other types of research data, focusing on the social sciences. Finally, we suggest areas where archives may need to revise policies, practices, and services in order to create secure, persistent, and usable collections of SMD.