"'S e do bheatha, Eairdsidh."
Translation:You are welcome, Archie.
To be more specific it is bheatha because it follows the possessive pronoun 'do'. As MK implies, nouns following the singular possessive pronouns mo, do and a (his), lenite, but not when 'a' means hers. The plural possessive pronouns - ar, ur, and an/am do not cause lenition. (I was going to ask MK why but see I would be wasting my time...)
Somewhere between Proto-Celtic, Primitive (or Archaic, or Proto-) Irish (the one written in ogham between 4th and 6th centuries) and Old Irish, lenition happened to consonants surrounded on both sides with vowels.
Initially it was a simple phonological phenomenon very similar to what happens today in Spanish – where b and v both are pronounced /b/ at the beginning of a word (banco /banko/ ‘bank’, vaca /baka/ ‘cow’) and after a nasal consonant (ambular /ambular/ ‘to walk’), but /v/ or /β/ in the middle (rebelar /revelar/ ‘to rebel’).
We don’t know exactly when it happened since in Primitive Irish it is not written (like Spanish doesn’t write it – there was no need, it was purely phonological thing, absolutely predictable).
But everywhere where lenition happens it is because some words ended in a vowel in Proto-Celtic and Primitive Irish, and thus made the consonant of the next word be lenited.
And so feminine nouns cause lenition of the following adjective, because most of them ended in -a in Proto-Celtic (similar to how most feminine nouns end in -a in Latin, Spanish, Slavic languages… all from the same Proto-Indo-European noun group).
mo was mo in Old Irish and *moy in Proto-Celtic, causes lenition because of final vowel (or semi-vowel), do was *do already in Proto-Celtic.
Singular a and plural an, am come from:
- masc. *esyo ‘his’ – ends in a vowel, causes lenition
- fem. *esyās – ends in consonant, no lenition, but historical lenited s appears as h-prothesis before vowels, eg. a h-athair her father
- plural *e(y)sōm or *e(y)som ‘their’ – which in OIr. was also a and caused eclipsis because of nasal m, and in Irish it is still so (eg. a dtoil ‘their will’), but Scottish lost eclipsis and restored nasal n, m at the end of this word.
All the plural forms, I believe, come from old plural genitive which always ended in -om or -ōm in Proto-Indo-European, hence eclipsis and no lenition in plural in Old and modern Irish (and no lenition in Sc. Gaelic).