An Argrathi cycle was equivalent to one Earth year. The crime of espionage required a minimum of fifteen cycles of correction on Argratha, and Miles O'Brien served 20 cycles, the equivalent of twenty Earth years (though his entire imprisonment was actually an implanted memory). (DS9: "Hard Time")
Cycles seemed to have a strong role in the religion of Borothan pilgrims. Amongst their religious concepts were the Cycle of renewal, the Cycle of creation and the idea that the universe went through cycles of rebirth.
The use of the word cycle as a "generic" alien unit of time is common in science fiction, to the point that it is almost a cliché. No doubt this was a factor to its many uses in the Star Trek universe.