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Yes, every man assigned to a ship has specific functions during various ship's events, including General Quarters--what you called battle stations. That roster used to be called a Watch, Quarter, and Station Bill (I don't know if they use the same terminology today).
Sadly (and my Research Services colleagues can correct me), the Watch Quarter and Station Bills were temporary documents, as the contents of the Bill changed frequently with the assignment of men to and from the ship--it would be impractical to keep all the copies of Bills as they changed, so none probably made it off the Little before she was sunk in her last action.
However, based on my extensive readings on World War II naval history, I would surmise that your father was assigned either to one of the antiaircraft gun crews or a damage control party. The ship had five single 5-inch 38 caliber gun mounts, each with a crew of 15, five twin 40mm gun mounts, each of which required 5-6 men per mount, and 7 20mm gun mounts, each of which required two men. So for a Fletcher-class destroyer such as the Little, well over 100 men served as part of a gun crew. As your father was a cook, his main job would have been in the galley, which would not have allowed him much time to practice gunnery, so during general quarters he probably passed ammunition to any of the 17 gun mounts on the ship. This was a critical role for the effectiveness of the antiaircraft fire from a ship is what kept her alive--the flow of ammunition to the gun mounts, especially those that fired rapidly such as the 40mm and 20mm mounts was very important.
If he was assigned to a damage control party, he would have spent general quarters inside the skin of the ship. Each damage control party had a number depending on where it was stationed in the ship, such as Repair 1, Repair 2, and so on. The damage control parties got their equipment ready for use during general quarters and stood by until they were needed in the part of the ship in which they were stationed. If some of the other DC parties were disabled by enemy action, a DC party from another part of the ship would move to the damaged compartments and begin their work of fighting fires, jettisoning ammunition, shoring up bulkheads, dewatering flooded spaces, and so on.
I hope you found this information helpful.