Now for the explanation.
When the Irish settled in New York, there was an island covered with rabbits. The word the Irish used for rabbits was coney, so the island became known as Coney Island.
The place became an amusement destination with a board walk and rides and everything. Someone got the idea of putting chili, cheese and onions on a hot dog one day. These sold well and several imitators came up and suddenly that was known as the Coney Island Hot Dog. After a few years the idea spread out, the name got shortened and it became the Coney Dog.
Incidentally, this is also how buffalo wings got started. A bar in Buffalo New York served very spicy chicken wings in order to get people to buy more beer. Other bars picked up on the idea and the wings became known for the town of Buffalo. They could give the wings out almost for free because chicken wings are possibly the most useless part of a butchered bird, used only for stocks and the like.
I just like food name etymology, I find it an interesting study.