I see how the relationship is reversed with Daidí Carraige and that makes sense in the direct translation. In fact the actual nick name of my father is papa rock. Even though he is my dad, I had always called him papa. Through prior research I had discovered their was not a translation to Gaelic for a word that means papa. I think people in general tend to look at papa as a name for a Grandfather. However, I did not like that translation for Grandfather. I didn't feel that it fit so I went with Daddy. The meaning of the nickname, Papa Rock/Daddy Rock, fits the "a father who is like a rock". The rock part meaning a strong/supportive person who was always there for you. I like the way Carraig d"Athair looks.

Carraig d’Aṫair

Does "athair" stand for father is in dad? Or does it stand as father as in a priest?

It can mean "father" in both senses, but it will be clear from context that it means "father" as in "dad".

If you were talking about a priest, you'd use a different word sagart.

I don't know if it is a mistake or just the way Carraig d'Athair would be written in old text but the script that Breandán wrote looks like it is missing the "h" in d'Athair?

I have used the Irish Font Writer from http://www.irishgaelictranslator.com/fontwriter/ and it also comes out slightly different in the traditional style.

No, it's not a mistake. In old script, the th is replace by a dotted t => ṫ.

Be very careful though, when I entered the translation into the font writer it came out as:

"Carraig d'/ Athair"

There is not suppose to be "/" in there.