So, why do we, as Christian non-ethnically-Jewish people, celebrate Chanukah (and other Biblical holidays, for that matter)? The answer is not actually all that difficult. It's surprising to me, in many ways, that we didn't do this a whole lot sooner:
So, we know that Jesus was Jewish, we know that God brought Jesus, the Messiah and savior of the world, TO the world through the Jewish people. These people were chosen and set apart as the people of the Promise, the people through whom redemption would come. And they were given a set of holidays and traditions that would honor God and keep them focused on serving and remembering the promises throughout the year. Because we, as Christians, serve that very same God, we choose to honor those holiday.
We also choose to honor the holidays that commemorate the times when the Jewish people have been miraculously and supernaturally preserved from destruction. We acknowledge that we serve a God whose covenant is everlasting and cannot be broken. Chanukah is one of these holidays. We believe that we, as believers in the Messiah and servants of the very same God, have been grafted in to the "family tree." In Romans 11 Paul says, speaking of the Gentiles (that's non-Jewish people):
"...you, although a wild olive shoot, were grafted in among the others and now share in the nourishing root of the olive tree..."
As a "grafted in" part of the tree we celebrate with the tree! We celebrate that the roots have been preserved and that we receive the nourishment of life and hope through this tree!