Eithne is pronounced ETH-na. This is the accepted pronounciation in English and in most Irish dialects.
Enya is the English spelling of the way Eithne/Ethna,etc can be pronounced in Donegal Irish. This dialect typically slurs and blurs constanants,for example Siobhán becomes "SHU-on".
Enya, the singer, is from the Donegal Gaeltacht of course and Irish is her first language.
It would be considered very affected if a non-Donegal person pronounced her name "
Enya".
Eithne is typically an middle aged/older person's name.I only knew one
younger Eithne.She was from Donegal and she said it "ETH-na" :)