This year, 2011, Hanukkah is from December 20-28.
Adam Sandler wrote "The Hanukkah Song" in response to the fact that there are so many Christmas songs (carols) and so few Hanukkah songs. But I've actually sung quite a few Hanukkah songs and Hebrew songs, both in and out of school.
Please review if you know any songs that aren't listed here! Happy holidays (to be politically correct)!
These songs are exclusively about Hanukkah:
"Happy Hanukkah, My Friend (The Hanukkah Song)" by Justin Wilde and Doug Konecky
"I Have a Little Dreidel"
"We Are Lights," lyrics by Steve Young, music by Stephen Swartz
These songs are a bit more general, Hebrew or Yiddish songs.
"Ma'oz Tzur," or "Rock of Ages"
"Yerushalayim Shel Zahav," or "Jerusalem of Gold"