Space information network (SIN) is an integrated network system of various space information platforms (e.g., satellites, stratospheric airships, manned or unmanned aerial vehicles) to enable real-time sensing, collection, transmission, and processing of various space information, as well as to realize both global and localized tailor-made systematized information services. Spectrum usage and management becomes a more and more serious issue in SIN mainly for the following reasons: (i) the paradox between spectrum shortage and spectrum under-utilization, (ii) the complex electromagnetic spectrum environment with tremendous spectrum devices and ubiquitous spectrum interference, and (iii) the spectrum disorder and spectrum attack. In this paper, we propose to empower SIN with big spectrum data analytics for dynamic spectrum sharing, real-time spectrum monitoring, and intelligent spectrum control. Specifically, we first identify critical spectrum issues in developing SIN and highlight that spectrum data analytics is the key solution to address these issues via spectrum sensing, spectrum data statistical inference and knowledge discovery, spectrum data-driven decision optimization, and spectrum experiment validation and evaluation, etc. Then, we introduce the concept of big spectrum data in SIN and analyze its characteristics, including volume, value, variety, viability, veracity, and velocity. Next, we discuss the emerging use cases and highlight research frontiers.