8 Reasons for 8 Days
Here are 8 approaches to answer this question:
They divided one nightís oil into eight portions. Miraculously, each portion lasted an entire night.
Beit Yosef, Orach Chaim 670
The Greeks ransacked the Temple many days in search of oil to defile. Despite their strength and numbers, they overlooked one flask. A few weak, battle-weary Jews found it immediately.
HaMeiri in Lehodos Uílehallel; Sefer HaEshkol, Chanukah 6:13
Wanting the oil to last, they made the wicks one-eighth of the normal thickness. Nevertheless, the flames burned just as brightly as if the wicks had been the normal thickness.
The golden Menorah was ritually impure. So were all the Jewish soldiers, having come in contact with death on the battlefield. Therefore, they were forced to make a temporary earthenware Menorah, because earthenware is more resistant to impurity. But earthenware is porous, and when itís new it absorbs a small but significant part of any oil put in it. Therefore, one nightís oil for a gold Menorah was not sufficient for an earthenware menorah because some of the oil is lost to absorption.
Bava Metzia 40a & Maharsha Chullin 55
In one account, the text reads "and there wasnít enough (oil) it to burn even one day..."
Sheiltos DeRav Achai Gaon, Parshas Vayishlach found in footnote to Megillas Antiochucus in Siddur Otzar Hatefilos
Seven days commemorate the miracle of the oil, and one day commemorates the miracle that a few weak Jewish soldiers defeated the mighty Greek legions.
Chanukah occurred in the year 3622 (139 BCE). Calendar calculations and other historical sources indicate that the 25th of Kislev, the first day of Chanukah, fell on Shabbat that year. Therefore, they needed to light the menorah before sunset Friday night, and consequently needed a little more than a nightís-worth of oil.
The commandment to light the Menorah with pure oil is written in the Torah (Leviticus, chapters 23 and 24) immediately after the commandment to observe the Succos festival for 8 days (7 days of Succos followed by Shemini Atzeres). The Sages saw this as a Divine hint that Chanukah should be for 8 days.
Bnei Yisaschar in the name of the Rokeach
Research based on Sefer Ner Lemeah, Rabbi Yerachmiel Zeltzer