While the melody is nearly ubiquitous among American Jews, it is by no means ancient, as Albert Stern explains:
Samuel E. Goldfarb penned “I Have a Little Dreidel” (with Samuel S. Grossman) [in the 1920s], while his older brother, Israel Goldfarb, composed [a widely used melody for the Sabbath hymn] “Shalom Aleichem.”. . .
The Goldfarb brothers grew up on the Lower East Side of New York in a family of eleven children that emigrated from Galicia. Samuel was born in 1891, and learned how to read and play music from his brother Israel, who was twelve years his senior. In 1914, Samuel—who was making music in Yiddish theaters and other popular venues—entered into an arranged marriage with Bella Horowitz, from the family that owned Horowitz-Margareten, renowned makers of matzah and Passover products. . . .
While Samuel started out playing piano in theaters, Israel—a graduate of the Institute of Musical Art (now the Juilliard School), the Jewish Theological Seminary, and Columbia University—rose to fame as a noted cantor, and later became the long-serving rabbi at the venerable Kane Street Synagogue in Brooklyn. . . .
“Generally speaking, in America Yiddish music influenced the popular music of Broadway and Hollywood,” said [Samuel Goldfarb’s son, Myron Gordon]. “With these kinds of songs, it was the opposite—it was an American tone being brought into a Jewish context.”
Read more on Times of Israel: http://www.timesofisrael.com/the-hidden-history-of-i-have-a-little-dreidel/