The riddle
What Do You Call a Turkey the Day After Thanksgiving?
A bird becomes a word
Ask what to call a turkey and most listeners reach for a bird name or a joke name. The answer treats the turkey as part of a meal instead, and that is what the day after Thanksgiving makes it. Merriam-Webster defines leftover as something remaining after a meal, and in the plural it is the usual word for the food itself. The word is plainly built from "left" and "over", which describes the situation exactly: what is left over once the dinner is done.
How long leftovers keep
The United States Department of Agriculture advises putting leftovers in the refrigerator within two hours of cooking, eating them within three to four days, and reheating them to 165 degrees Fahrenheit. By that guide, a turkey held for the day after the holiday sits comfortably inside the safe window, with a few days to spare before the platter should be emptied.
Why the day after matters
The riddle specifies the day after because the label only fits once the meal is over. On the day itself the same food is dinner. The next morning it is a different category, even though nothing about the turkey has changed. Many riddles work this way, by moving a familiar thing into a new frame and letting the word for the frame do the joke.
Naming the plural
"Leftovers" is almost always plural. A single leftover exists as a word, but people rarely say it about a meal. Telling the riddle works best with a flat voice and a straight face on the last word.