According to Greg Weisman, while Cu Chullain was protecting Ulster in the place of the "Hound" (a gargoyle beast) he originally killed, he would also be raising and training a new "Hound" (another gargoyle beast) to eventually take his place.
It was remembered in Ireland long after Cu Chullain's death, and even by the 1990s, tales were still told about the Great Beast; it was said that to hear it baying meant trouble, and to see it was even worse. The Banshee mistook Bronx for the Hound of Ulster when he arrived in Ireland on the Avalon World Tour. ("The Hound of Ulster")
Real World Background
In the original Cu Chullain legends, the Hound of Ulster was actually Cu Chullain's title; it was ascribed to him because of his faithful defense of Ulster, the kingdom of his uncle King Conchobar, and as an echo of his name "Cu Chullain", meaning "Hound of Culainn". In this sense, the Hound of Ulster is an invention of Gargoyles.
However, it also echoes the actual legends about "black dogs" found in the British Isles, who were said to serve as portents of doom. (Indeed, in the original plans for "The Hound of Ulster", this beast would have also been called a "Barghest", an actual "black dog"-like creature in the folklore of northern England.)
Their howling was said to serve as a death-omen, and sometimes the sight of them was horrible enough to kill whoever saw one. (The Hound of the Baskervilles, whom Sir Arthur Conan Doyle created for his Sherlock Holmes stories, is a reflection of such legends. And Elisa Maza mentions Holmes in the episode.) However, at times the "black dogs" would serve as protectors, walking alongside travelers late at night and keeping them safe from harm.
Obviously, in the Gargoyles Universe, such tales about "black dogs" must have originated from gargoyle beasts.
Both traditions about them fit the nature of gargates; the stories about them as menacing beings reflect the fear and suspicion that humans have towards gargoyles and gargoyle beasts, while the "guardian black dog" stories reflect the true nature of gargates as protectors.