Waits explains the origin in an interview:

The Eyeball Kid is a comic-book character. Actually, it was Nic Cage that reintroduced me to comic books. I hadn’t thought about comic books since I was a little kid, but he seemed to carry that mythology with him. It was inspiring to see him keep alive some of those principles that we associate with childhood, to the point where he named himself after Cage, the comic-book hero. But I was trying to imagine what it would be like for a person with an enormous eyeball for a head to be in show business. If Barnum & Bailey were still around, I imagine he would have thrown in with them. The tour would be sponsored by Visine or Bausch & Lomb. It’s a metaphor for people that get into show business, because they usually have some kind of family disturbance or are damaged in some way or another. I had a manager when I was a kid, I threw in with a guy named Herbie Cohen, who worked with Zappa. I wanted a big bruiser, the tough guy in the neighborhood, and I got it.