1550s, "one of the ancient sect of philosophy founded by Antisthenes," from Latinized form of Greek kynikos "a follower of Antisthenes," literally "dog-like," from kyōn (genitive kynos) "dog" (from PIE root *kwon- "dog").
Supposedly the name is a reference to the coarseness of life and sneering surliness of the philosophers, and the popular association even in ancient times was "dog-like" (Lucian has kyniskos "a little cynic," literally "puppy").
But more likely it is from Kynosarge "The Gray Dog," the name of the gymnasium outside ancient Athens (for the use of those who were not pure Athenians) where Antisthenes (a pupil of Socrates), taught. Diogenes was the most famous. Meaning "sneering sarcastic person" is from 1590s. As an adjective from 1630s.
[Diogenes] studied philosophy under Antisthenes, a crusty type who hated students, emphasized self-knowledge, discipline, and restraint, and held forth at a gymnasium named The Silver Hound in the old garden district outside the city. It was open to foreigners and the lower classes, and thus to Diogenes. Wits of the time made a joke of its name, calling its members stray dogs, hence cynic (doglike), a label that Diogenes made into literal fact, living with a pack of stray dogs, homeless except for a tub in which he slept. He was the Athenian Thoreau. [Guy Davenport, "Seven Greeks"]