This concept works best if you play in person, but could possibly work in online games as well. Your deity becomes the cat that lives with the game host. Because cats react to humans but are not as easily trained as dogs, cats work great as stand-in deities, just as aloof as canon gods.
Are you trying to use Divine Intervention? First, you have to try and get the cat to come over to you. Then, what your god does if you manage to summon it is not up to you, it depends on what the cat does. Does it chew on a piece of terrain? Some part of the architecture is going to become unstable and collapse. Did it knock over a miniature? Your god has smote that foe or ally. Did the cat bat some dice around? You just gained advantage for 1 minute. Did the cat purr or show you affection? You're healed by some amount and gain a use of Inspiration. Did the cat hiss or run away from you? The god you are trying to reach is unavailable and will have to call you back in 1-5 business days.
Because neither the player nor the DM is in full control of your god, you never really know what getting their attention is going to do.
Arcana: Especially for black cats, cats have long been tied to witchcraft and sorcery, as well as associations with demons, faeries, and certain deities. So, them having an Arcana Domain is fitting. Some deities associated with cats, such as Freyja, are also associated with magic, while others, such as Bastet, are associated with protecting people from evil spirits and diseases, both of which lend credence to the idea of a mystical cat deity.
Death/Grave: Similarly, cats have long been associated with death. Sometimes omens of misfortune, other times psychopomps that lead others to the after life. The persecution of cats as agents of devils or witches in the middle ages may have contributed to the unchecked rodent populations in Europe having an impact on the Bubonic Plague. Cats are also very skilled hunters, they just happen to be the same size as human babies. Introducing cats to ecosystems has wiped out or endangered entire rodent/bird populations as a result.
Trickery: It's hard to know what a cat is thinking at any given time unless you learn how to read their body language. To many, cats are enigmas. Asking for attention one second then biting and scratching the next. Cats have long been associated in folklore with trickster archetypes, thieves, spies, demons, and faeries. Cats are also seen variously as symbols of good or bad luck, and gods of luck fall under the umbrella of the Trickery Domain.
Twilight: Cats are primarily nocturnal creatures, and also sleep 18 hours a day. The twilight domain does not solely focus on darkness, the moon, nighttime, or the stars. The twilight domain also encompasses gods of sleep and dreams, and those who safeguard homes and communities against whatever lurks beyond the lantern glow of the village, fending off the dangers that lurk in the night to protect those at rest. Cats are extremely community-focused creatures that share resources when they have successfully hunted, will bond with and mimic other animals if raised with them, and care for sick or elderly members of their clan.
Of course, the unknowable and mercurial nature of cats also makes them work well as Warlock Patrons, perhaps even better than as gods, since their behavior can be hard to predict. Even sweet cats sometimes bite for little to no reason. Even naughty cats purr. I had the idea that the cat's nature would impact the patron they count as: