I haven't read all the responses and don't know if you've come up with something, but here's my story.
I suspected that my DGD would most likely be calling the other grandmother "Grandma" since she lived much closer. But she wouldn't make the "G" sound, so she would say "Mommy" for her mother and "Mama" for her other grandmother, and wouldn't call me anything. So we finally got her to say "Mimi" and that became my name. I did not want her to use my first name "Grandma Diane". When we finally settled on "Mimi" it was apparent that she was as pleased as I was that she had a name for me!
What nationality are you? Is there a term for Grandma in a different language that you like? My other DD married a German, so I'm sure the other grandmother (when they have children) will have the German term (whatever that is). I'd like to stick with Mimi, so all the grandchildren call me the same thing.
Congratulations BTW!