My 7-year old daughter does this with the fish (who, ostensibly, are “hers”), she recycles the name, but mainly because she likes the name (her favourite stuffed toy has the same name).
Since fish are the Bic lighters of pets, I don’t stop her at all …
My Jamaican landlady (who definitely fit the “little old lady” definition) called her cats pussy, which always stroke me as particularly funny given that several of them were toms. What totally cracked me up is that she would look both ways and get this “I’m being a naughty girl and I know it!” look in her face before saying “kitty.” I figured it was a dialectal variation and confirmed it with her best friend.
Recycling names within a family, or naming a child after a (possibly deceased) relative, is different from naming a pet after your own dead child. Most people have the sense that a human is worth more than an animal, and to be blunt about it, the cat’s going to die in 10 or 15 or 20 years and you may wonder what she’s going to do then - keep naming cats after him?
If my parents ever did something like this I would be very uncomfortable. But there’s not much to be gained from giving her flack about it. I’d make sure to keep an eye on her to see if she’s having trouble, but she’s probably not in the deepest stages of grief or anything.
And hey, in time she might feel differently about this. It’s not like you can’t change pet’s name.
I agree with calling the cat your own nickname. If it’s a cute nickname, and you do it enough, it may carry over to your mother eventually. Our cat’s name has gradually changed from “Cocoa” to “Fatboy” simply through the persistence of my daughter (the cat’s not even fat).