Mí means month. the correct answer if they want what asked for-June-would be Meitheamh. They did not ask for Month of June (mí an Mheitheamh), therefore Duolingo is wrong because Meitheamh is not provided as an option. There are not two correct options-there are NO correct options.
It's not meant to be translated literally. For instance, my basic Irish textbook puts it this way:
In everyday speech, people tend to use the word mí (month) when referring to a particular month:
Beidh an bhainis ar siúl i mí Iúil.
The wedding will take place in July.
They're really two ways of saying just the name of the month, not literally "month of [month]".
That said, the multiple-choice and word bank selections can be completely ridiculous at times, so I absolutely get the frustration - but please remember, these are generated automatically, and the course content creators are not able to affect them.
As far as I know, if 'mh' is slender, it is always pronounced /v/ regardless of dialect. Also, the 'i' in 'ei' is always silent. So, this would sound like /vehiv/.
(EDIT: I just noticed this, but I think the standard answer used to be the form Mí an Mheithimh. Mheithimh is what I was describing.)
I got this as a 'pick from a list of words' question which only had 'mheithimh' as a 'June' option, so it obviously marked this choice as wrong, but I had no option. There wasn't an appropriate option to report this kind of problem in the 'report it' option list either, but that's got longer since last I looked at it. (This problem occurred Oct 2018.)
Update - at the end of my session I obviously got the same question repeated, but this time also got mí and an as options so Mheithimh was accepted when combined with these. The problem then is the absence of Meitheamh, mí & an in the first presentation; and just the absence of Meitheamh in the second.
I took a look in the admin system, and the only accepted options are Meitheamh and Mí an Mheithimh - meaning that you suffered a bug, as did everybody else who has submitted one of the many error reports. Since I'm not part of the Irish course (I'm on the Swedish team), I can only look, not change anything... but since it seems to be a bug, it's probably not something which the course content creators can affect anyway.