While not listed among the Feasts of The Lord, Hannukah is mentioned in the Gospel of John (10:22-23) as the Feast of Dedication.
Hannukah celebrates the defeat of a wicked Hellenistic ruler, Anticochus Epiphanies, who defiled the Temple around 167 B.C.E. by sacrificing a pig on the alter and then setting up an image of himself on the altar. This was done to try to force the circumcision, and to accept the Hellenism. This action only galvanized a few stalwart Jews and priests to fight back.
After a three year battle, the Jews won against all odds. On the 25th of Kislev, the Temple was rededicated and the Menorah that symbolized the light of The Lord was relit. But then this revealed a new problem--there was only enough of the special lamp oil to last one day. They decided it was better to light the Menorah anyway. To their amazement, the light stayed lit for eight days, allowing enough time for new oil to be prepared. The Temple was restored and rededicated to the glory of G-d and an eight - day festival was established. Thus the reason today for Hannukah to be celebrated for eight days and lighting candles every night during that time.
This story is told in the Book of Maccabees.
Why should Christians celebrate Hannukah? Because Yeshua did! Not only did he celebrate it, but he observed it in the same Temple that had been cleansed and rededicated just a few generations earlier!
Hannukah is a reminder of the faithfulness of God. Satan, through Anitochus had planned to destroy God's word and His people through assimilation and annihilation. Had he been successful, there would have been no more Jewish people, no Messiah to come, and no Calvary. Men and women would have been lost forever in sin, without hope.
Therefore, Hannukah is a celebration of many things. Christians can remember God's faithfulness not only because he delivered His people, Israel, from Anitochus; but because He was faithful and sent His son, our Jewish Messiah, to deliver people from their sin.
Finally, Hannukah is a reminder of the importance of not allowing oneself to be "assimilated" into the world. This was part of the plan that Anticochus was implementing to eliminate the Jewish people. This same plan is still in effect today for Jews and for Christians.