This case contained the right headband during the sleep experiment on-board the Shuttle-Mir.
The night headband system was part of an experiment to monitor eye and head movements while an astronaut slept to understand better the quality of sleep in weightlessness. The headband was wired to collect data from the attached T-shaped sensors that fit on the person's upper eyelids. Data flowed to a small monitor worn in an armband.
This object is not on display at the National Air and Space Museum. It is either on loan or in storage.