If she pronounces it like my mom's cousin did, then "Eithne" is pronounced "Ethna". Mom's cousin (actually cousin in law) grew up in Ireland and that is her name. It is the gallic spelling of the name, I think. Of course, I'm not the originial poster with that name so I could be completely wrong, and if so, I'm sorry to have jumped in.
It is Eth as in Ethel and Ne like in Never although the official Gaelic pronunciation is Enya but no one gets that from Eithne (which is why she changed her name to that spelling) so my mum decided to go for a somewhat phonetic pronunciation