Blueskies - my daughter was the oldest of my Gran's great-grandchildren and, whilst I was pregnant, there was a lot of pressure on me to "get" my daughter to call her "GeeGee" (for "Great-Grandma"). It didn't work. My daughter simply called her "Grandma", like I did when I was growing up. Then again, my grandmother was a bit... forthright in her choice of what she would respond to by grandchildren ("I insisted that you call me 'Grandma', not 'Nanny', because I am not the hired help!", being just one of the arguments as to why I, as a 16 year old, was expected to call her "Grandma" and not "Gran" - which she thought was "too common"...
). I took to calling her Gran eventually, and she acquiesced... but by that time, there were 6 grandchildren and one great-grandchild, so maybe she'd just given up and learned how to go with the flow...?
On the other side, I had grandparents and great-grandparents who were all expecting to be called the exact same name. When I was 2 or 3, I got a bit fed up of calling "Nanny" and two women bickering over whether I meant them or not, whilst ignoring me... so I took to using the endearing terms of "Big" and "Little" to differentiate. Everyone assumed, still to this day some 40 years later, that it was a generational "thing" - that "Big" meant "great", which yes; yes it did... (except actually, my great grandmother was a bit on the plump side, and her daughter, my grandmother? Was all bones and angles. Apparently, I was very sizeist when I was a toddler...). My grandfathers were both of the same build, but were tagged by their wife's moniker. It worked. All of my younger cousins followed the trend, and now? My "Little Nan" is "Big Nan" as far as my children are concerned, even though she's still all elbows and angles!
Children will name as they see fit. If I have grandchildren, I hope they'll call me something that I won't mind answering to in public... but other than that? It's beyond my control. Although I'm thinking "Granny Green Hair" or perhaps "Granny 'Be Kind (To One Another)'". 