Thanks to a beautiful coincidence of calendars this year, the Jewish festival of Chanukah 5777 and the Christian feast of Christmas 2016 both began on the same night. And, since each is celebrated as an octave, both will continue through New Year's Day 2017. Of course, unlike Easter and Pentecost which are connected with the Jewish festivals of Passover and Shavuot both historically and theologically, no such links connect Christmas and Chanukah. Jesus may well have been born in December, but he could just as well have been born at any other time of the year, and the Christmas story has no symbolic associations with the Chanukah story.
The Gospel of John likes to call attention to Jewish festivals, and Chanukah does get a passing reference in John 10:22. But, unlike the greater Jewish festivals, Chanukah imagery plays no role in the New Testament's presentation and interpretation of Jesus' life, death, and resurrection.
Of course, both Chanukah and Christmas are winter holidays, and many of our Christmas customs are really winter-related, festival-of-light customs. So a sort of connection can be made at that level. But I think we can take the connection even a little further. Chanukah commemorates the rededication of the Jerusalem Temple after its desecration by the Seleucid King Antiochus IV (cf. 1 Maccabees 4:52-59). Historically, it recalls a great Maccabean military victory, which, while a great and heroic national accomplishment, was itself certainly seen as somewhat miraculous. But the actual festival, as actually celebrated in Jewish religious practice, focuses on another miracle, one which is even more obviously an exclusively miraculous, entirely divine accomplishment.
God's intervention in history to save his people and God's abiding presence with his people (signified by the Temple and, more immediately, by the miracle of the oil) are Chanukah themes which resonate with the Christmas story. For the Incarnation is, after all, the fulfillment of God's ancient promise to save his people, which he does by becoming present among us in the ultimate Temple that is his Son's humanity. And like the miracle of the oil, the Incarnation has a modest, ordinary appearance.
Just as the Chanukah story contrasts the monstrous pomposity of the Hellenistic king and his cosmopolitan collaborators with the more powerful simplicity of God's miracle of the oil, so too the Christmas story contrasts the the false power of pagan emperors, kings, and governors (Augustus, Herod, Quirinius) with the true power of the Word-made-flesh.
And just as, despite human and earthly obstacles, the divine presence continued in Israel as symbolized by the long-burning oil, so too God's great visitation of his people continues in the incarnate Word now present forever in his Church.