Billy the Kitten is one of the “Rare Cats” in Neko Atsume. The way to attract a “Rare Cat” to your Neko Atsume yard/home is to place an item that a particular special cat likes. Eventually, the cat you are trying to convince to visit you will appear.
Billy the Kitten only likes one item – the Cowboy Hat. You can buy it in the Neko Atsume Shop for 400 silver fish. Several other Neko Atsume cats also like to use the Cowboy Hat.
The Cowboy Hat is described as “This relic of the Wild West will shade your cat’s eyes on the range.” I put the Cowboy Hat next to a food dish. You can place it in any of the spots that the game allows players to put items. The screenshot below shows what the Cowboy Hat looks like when no cats are using it.
When Billy the Kitten visits, he sits next to the Cowboy Hat. As you can see, he has his own Cowboy Hat to wear. Several other Neko Assume cats will use the Cowboy Hat. They seem to use it in one of two ways. Some will try and wear the Cowboy Hat. Others like to sit inside the Cowboy hat. It is as though they are thinking: “If I fits, I sits.”