Eithne,Ethna and Ethnne and other spelling variants are generally pronounced Ethna in Ireland. It is rather out of date name and you don't meet many younger Eithnes.
However a confusion has arisen among non-Irish speakers and people who live outside Ireland about the pronounciation of this name. This is because a lot of people first encounter it through the singer Enya. En-ya is a regional,minority pronounciation of Eithne. The singer is a native speaker(Irish as first language speaker)from Donegal. The most notable effect of this dialect is a slurring of constanents. Thus Máire becomes Moya, Eithne becomes Enya,etc. However not all Eithnes from Donegal pronounce it Enya. The only Eithne I've met under 50 is "Eth-na".
I have noticed in recent years a popularity in Enya as a given name.