What connections does Chanuka have with Succos?
- Both are celebrated for eight days
- Both have full hallel on all the days
- Both are celebrated outside the house (sukkah is outside, and menorah should be outside)
- Both are Possul if they are more than 20 amos high (i.e. above the ground)
- Both call for hiddur mitzvah (lulav we try to get the best, and menorah by adding more each night)
all taken from here.
The opinion of Beis Shamai that we should lit candles in a decreasing way (from 8 to 1) has something common with the fact that Korbanos of the Succos are also decreasing (from 13 to 7).
When the Chashmonaim rededicated the Beis Hamikdash on the 25th of Kislev, they celebrated for the next eight days with lulavim (symbolically making up for having missed that mitzvah in its proper time because they were in hiding).
(II Macc. 10:6-7)
(an expansion on earlier answers)The Beis Yosef's question about why we celebrate eight days, when the miraculous long-burning oil only lasted seven days more than it normally would.
The Sefer Chashmona'i says that in the year before Matisyahu's rebellion, Antiochus prevented the korbanos of Sukkos and Shemini Atzeres. Therefore, the Jews were able to re-inaugurate the Beis Hamikdash, they celebrated for eight days, in order to show that they were making up for the lost days of Sukkos and Shemini Atzeres.
In the book of macabbees chanuka is called "sukkot b'kislev"
Halichot Teiman page 65 - After Chanukah, children excitedly scatter in the streets, knocking on people's doors and shouting "thebil!" (Arabic for "wicks!"), and people give them the wicks and the oil left in the Menorahs. The children gather as much as they can and give it to their teacher, who makes it into a healing ointment, by mixing it with ground-up myrtle leaves (Hadassim) that were used for the Mitzvah (of four species during Succos). This (ointment) is subsequently given to anyone who suffers from minor infected wounds.
Source: Mishpacha Kolmus #15 Chanuka 5771