Mementos show they care
Each cat will bring you a special little gift at some point, maybe it will be a freshly laundered handkerchief or a Cicada skin. It is each cat's way of saying thank you.
The way to get a cat to bring you a Memento is to get it to visit a lot. They will need to visit anywhere from 10 to 15 times before they will bring you something (though I've heard that Sapphire has been known to bring a memento after her first visit and Rascal takes at least 20 visits). If you notice any cats that haven't brought you a Memento yet, look at their page in the Catbook and place one or more of the Goodies on their top three list in your yard to ensure they appear more often.
To see which Mementos you already have, tap Menu, Cats, and then Mementos.
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