Kislev is the month of Chanukah--the only holiday in the Jewish calender which spans, and hence connects, two months: Chanukah begins on the 25th day of the month of Kislev and concludes in the month of Tevet (either on the 2nd or 3rd, depending on the number of days in Kislev).
The name Kislev derives from the Hebrew word for "security" and "trust." There are two states of trust, one active and one passive, both of which are manifest in the month of Kislev (see Bitachon, confidence). The miracle of Chanukah reflects the active trust of the Chashmonaim (Maacabim) to stand up and fight against the Hellenistic empire (and its culture). Kislev's sense of sleep reflects the passive trust that G-d's providence always guards over Israel.