I think your view of yourself can change over time. For example, I grew up never having my name shortened, and when I did my nurse training, there was another girl with the same name, who did use the shortened version, so she was, let's say, Cathy, whilst I was Catherine. When I left nursing and went to university, I decided new path, new me, new name, and introduced myself to everyone as Cathy, and have been Cathy ever since - it felt right at the time, and still does.
That said, I do appreciate that it would be very difficult for those who knew me before the change to start calling me Cathy, so I know I am Catherine to them, and Cathy to everyone else, and I don't let that bother me.
Maybe your MIL feels Gran is more 'her' than Grandma, and that's why she would like to change. But she will have to understand that it is not easy for other people to make that change, especially if they are small children.
You can explain to her that people can shorten their names, and titles - and that whilst little children call their parents mummy and daddy, when they are older, they call them mum and dad, and that this is something similar, and she isn't losing her marbles.