It's funny that you used An Cheathrú Rua in your example, a Bhríd, because that's where I have my Islanders originally coming from. Small world.

Cool !

I hope there is a character named Bríd incorporated into the scéal. She could be a woman of ill repute causing all the scandal in the area!

We have to use surnames officially.
But locally most native Conamara people will be known from their parents/grandparents first names (It's probably the same in other Gaeltachts).
You could come to a town like An Cheathrú Rua, for example, and ask for Bríd Flaherty. You'd be asked which one out of the 20 Bríd Flahertys do you want. (in rural areas houses don't have numbers either). But ask for Bríd Bhaba Chóilín and people will know right away, that name makes you unique.
People that are known only by their surnames are usually blown -ins, who have no roots in the area.

So, Bhaba and Chóilín are your parents' first names?

No. You follow the one family line. Either your mother's side or your father's side.
It more often the father's side. Or you can be known as both depending on which community you came from. (After you marry you can also be known from your husband's line as well)

So Baba would be my mother, and Cóilín my grandfather, Seán my great-grandfather, Antaine my great-greatgrandfather etc etc.

I am Baba's daughter, so the H goes in to make it genitive.

Bríd Bhaba Chóilín Sheáin Antaine

(two generations are usually enough to identify you)

You follow the one family line. Either your mother's side or your father's side.

Maybe the one that is best known in the community.