The name most likely refers to Troy, New York ("Ilium" was the name the Romans gave to ancient Troy). In all other respects, Ilium very closely resembles Schenectady, New York, with the fictional Iroquois river standing in for the real Mohawk River, which flows west-east through Schenectady. The Ilium Works is in roughly the same geographic location as the General Electric plant in Schenectady, where Vonnegut worked as a public relations writer. Still, the city of Ilium is quite clearly distinct from Schenectady, as characters in Player Piano, Cat's Cradle, and Slaughterhouse-Five refer to Schenectady as a separate place.
In Vonnegut's Galápagos, the character of Mary Hepburn was a High School teacher in Ilium, and in Cat's Cradle, it is the former home of Dr. Felix Hoenikker - one of the fathers of the atomic bomb - and, thus, the town that Jonah visits to interview Dr. Asa Breed, Hoenikker's former supervisor.