The rubber ducky room was a room aboard the USS Cerritos that was considered off-limits.
In 2381, Ensign D'Vana Tendi thought she was being transferred off the USS Cerritos and visited parts of the ship she wanted to remember. When Sam Rutherford noted he had an idea of somewhere off-limits, Tendi asked if he was suggesting the "rubber ducky room", but he thought the captain's yacht was "even better." (LD: "First First Contact")
The rubber ducky room was an in-joke referencing an icon of a rubber duck included on the USS Enterprise-D master situation display, which was itself a joke that was never meant to be legible on screen.