A hat box was an object for carrying hats.
The Fourth Doctor jokingly told George Litefoot that Leela was found floating down the Amazon in a hat box. (TV: The Talons of Weng-Chiang) Leela later told Ellie Higson that she had lost her luggage in the same manner. (AUDIO: Dead Men's Tales)
Donna Noble had a hat box prepared on the off chance she would meet the Tenth Doctor again. She did and handed it to him to carry, saying she was ready for the "Planet of the Hats". (TV: Partners in Crime)