Mr. Meowgi is a Rare Cat in Neko Atsume. The way to attract a Rare Cat to put the things that the Rare Cat likes into your yard/home. Eventually, the cat you are trying to collect will appear. Several other cats will also use the special item.
Mr. Meowgi likes two items: the Sakura Pillow and the Scratching Log. It took a very long time before Mr. Meowgi sat on the Sakura Pillow, but he used the Scratching log almost right away.