I don't mind anyone using my first name (as long as they ask me if I mind) because my surname is Greek and a lot of people seem to find it difficult to say - I can't stand hearing it mispronounced, which they do, even when I pronounce it carefully for them.
My daughter's friends always called me by my first name, which I didn't mind at all, even when they were tiny. When I first met my American son-in-law I told him to call me by my first name, but not long before the wedding 18 months ago, he lost his own dear Mum and as soon as he got married to our daughter (at the wedding actually), he asked me if he could call me Mum now, so, of course, I said Yes!
My husband's daughter, Alexandra,who is half Norwegian and half Greek and lives in Denmark as she is married to a Dane, has two dear little children aged nearly three and seven months and we had to decide what they were going to call me as they have two grannies already (luckily, we all get on like a house on fire!) In Scandinavia your Father's Mother is Farmor and your Mother's Mother is Mormor and they were used up already, so they asked me what I would be called by any English grandchildren and I plumped for Nana, which is what I called my grandmother and our daughter called my mother. So that is what they are going to call me!