This is a question I ran into in Mozambique, where the national language is Portuguese, a few years ago. Something I read today reminded me of this, and since I know there are some Portuguese speakers here I thought I'd ask about it.
Mãezinha means mommy, more or less--mãe meaning mother, and zinha being the diminutive ending.
In Mozambique, I came across a number of little children called Mãezinha. I finally asked a woman why, and she told me it is a nickname that is given when a girl child is named after her grandmother. She told me that just as it's disrespectful to call your mother by her first name, it would be disrespectful to run around after your child calling her by your mother's first name. So the family calls such children "Mãe" or "Mãezinha" out of respect to the grandmother she's named after. The woman I spoke with was rather astounded that we would be crass enough to call our children our mothers' names and use that name in everyday life.
I never ran into any children called "Pai" or "Paizinho," so as far as I know the custom doesn't apply to male children, but I could be wrong about that, my experience was rather limited.
Anyway, I thought that was a rather interesting and sweet custom, one I'd never encountered in Portugal, where I learned Portuguese. I was just wondering if this is strictly an African custom, or if any Portuguese speakers are familiar with it from other places? Is it perhaps prevalent in other languages/cultures and I'd just never heard of it before?