"His name and her name."
Translation:A ainm agus a hainm.
Lenition and h-prosthesis are two distinct initial mutations, and they each have their own rules. For example, verbs are lenited in the past tense (caitheann sé vs chaith sé), but verbs that start with a vowel are marked with d' (ólann sé vs d'ól sé). If "h-prosthesis" was just "lenition for vowels", you'd have h-ól sé.
Lenition and eclipsis don't apply to vowels. When it comes to figuring out how to mutate words that start with vowels, just be aware that they don't follow the same rules as words that start with consonants.
a meaning "his" causes lenition. Lenition is a process that only applies to consonants.
a meaning "her" causes a h-prefix to words that start with a vowel. A h-prefix is not lenition.
a sheanmháthair agus a seanmháthair - "his grandmother and her grandmother"
a ainm agus a hainm - "his name and her name"
a cat agus a cat - "her cat and her cat"