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.