Great-grandmother (95) wants another dog. How do I convince her she does not need anymore animals?

You could tell her there is a rule about how many pets you can have in your area. Or in your house...LOL

a person who surrounds themselves with animals is , by clinical description , not always firm in the head . dont get me wrong . i dont give a shat , but ill tell you one thing . as of 7 am this morning i had one mouse up here who thought we were bff ' s . i caught him napping after whacking my leftover sweet rice and i skewed him with my dental pick .
thats where my head is at ..

You could try telling her that it wouldn't be fair to the dog, even an adult dog is probably going to outlive her.

You just say "NO" without sounding angry. Kindly, sweetly, kiss her cheek and say NO.