The eighth day of Chanukkah has a special meaning as the culmination of the holiday.
The number 7 represents the week, Shabbat, the sabbatical years, the omer (7X7).
The number 8, which is equal to 7 + 1, is a day after things were completed. 8 is therefore beyond time, it is eternity. The eighth day is the essence of Chanukkah and the remembrance of the Light that is forever present in the world.
(Source: summary from the book "The Jewish Festivals" by Michael Strassfeld)