It's a custom, writes R' David Seidenberg here at NeoHasid, to meditate on the light of the Chanukah candles and to sing while doing so. He offers a beautiful niggun for slow, quiet Chanukah meditation. The melody is a Skolyer one; the words are the words to Ana B'Koach, which I posted about back in August. (If you want to hear him sing it, that NeoHasid page is the place to go.) While Chanukah isn't as rife with melody as some other holidays of this time of year, there are melodies which suit the season. Here are a few.
One of the songs that's traditionally sung after the candles are lit is Hanerot Hallalu, which talks about how we kindle these lights in remembrance of the miracles and wonders which God did for our ancestors at this season and how the lights must be kindled only in order to spark our mindfulness, not to be used for mundane purposes. You can hear an elderly Israeli gentleman chanting that one, after the other Chanukah blessings, at the end of this short video on YouTube here. Or, for something a bit more musical, here's a recording of women's voices singing an adaptation of the prayer in harmony, in Hebrew and in English.
I remember standing in the dining room of the house where we lived when I was a kid, singing selected verses of Maoz Tzur, "Rock of Ages," in Hebrew and in English. It turns out that the song was written about 900 years ago, and the tune that I grew up on is an adaptation of a German folk song which was also favored by Martin Luther when he chose choral settings. (Read all about it at JHOM: About Maoz Tzur (Rock of Ages).) If the classical translation by Gottheil and Jastrow isn't your style, you might dig Reb Zalman's singable translation. Anyway, here's the tune I grew up singing [mp3], performed by a choir accompanied by piano.
One of my favorite songs to listen to (or sing) during Chanukah is "Or Zarua" by Shir Yaakov. (You can listen to it here at duck.fm.) It's not a Chanukah song per se, but it feels thematically appropriate to me. The words translate to "Light is sown for the righteous / and for the upright of heart, joy," and the second line is "For with You is the source of life / in Your light we see light." The theme of light in the darkness is a resonant one at this season. (Once again I apologize to readers in the southern hemisphere, for whom all of this winter/darkness material couldn't be more wrong.)
If you're a fan of folk music, you probably already know "Light One Candle," Peter Paul & Mary's paean to the Maccabees, to the "terrible sacrifice justice and freedom demand," and to the "wisdom to know when the peacemakers' time is at hand." Anyway, you can watch it here on YouTube. (I think their retelling of the story of the Maccabbees is necessarily limited, but that doesn't make it any less of a great song. If you want discourses on the history and implications of the holiday, try R' Arthur Waskow's Origins of Hanukkah: Miracle or Revolt? or my post Mai Chanukah? But I like the PP&M song just as a song, even if the history it offers is a bit redacted.)
On a different note, I can't resist offering this link again, though I know I posted it last year too: Hallel using Christmas carol melodies. It's customary to sing the psalms of Hallel every day during Chanukah; if you like Christmas tunes and you get a kick out of the idea of using secular (or other traditions') melodies for celebratory psalm-singing, then you might dig this too!
I made a solstice playlist a couple of years ago as the days were growing shorter and darker. I had a fantasy of sharing all of the tracks here, but many of my favorites don't seem to be online anywhere -- Richard Shindell's "Darkness, Darkness," Richard Thompson's live rendition of "Remember O Thou Man," Maria Kalaniemi and Sven Ahlback's "Randas" (which means "Dawning.") Even the Barenaked Ladies' "Hanukkah Blessings" song, which features their own lyrics alongside the words of the Chanukah blessings, isn't online anywhere! So instead I'll ask: what music is enlivening your Chanukah this year?