Eithne, Ethna and Ethnne and other spelling variants are generally pronounced ETH-na in Ireland.
However a confusion has arisen among non-Irish speakers and people who live outside Ireland about the pronunciation of this name. This is because a lot of people first encounter it through the singer Enya. En-ya is a regional, minority pronunciation 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 consonants. 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" and is from County Donegal. [noted -ed]
ETH-na is the English pronunciation. The Irish language does not have the "th" sound. (Indeed, a good many Irish have trouble pronouncing the sound in English, as is well known.)
In the Irish language, th is pronounced like English h, though it is usually dropped at the end of a word. In this case, it may or may not be dropped, depending on dialect.