Tonia Barrows wearing a hennin
Kira and Dax wearing hennins
A hennin was a tall, conical shaped hat worn by European noblewomen, which dated back to Earth's 15th century.
Yeoman Tonia Barrows found a hennin on the Shore Leave Planet. Upon finding it, she stated, "a girl should be dressed like a fairy tale princess, with lots of floaty stuff and a tall hat with a veil." (TOS: "Shore Leave")
When Lieutenant Commander Worf was introduced to Major Kira Nerys and Lieutenant Commander Jadzia Dax, they had just emerged from a holosuite wearing these hats, prompting Worf to compliment "nice hat," and the officers to clarify that these were not part of their usual attire. (DS9: "The Way of the Warrior")
The hat was not referred to by this name in these episodes. It was merely called a "tall hat" by Barrows.