I think "He goes out with a new head" should be accepted. It's a little idomatic, perhaps, but correct nonetheless.
Well he would be a male who is going out with a new male friend, "a head" (inf. Hiberno-English) or with a new outlook on life.
How do we know the new friend is male? What would the sentence be if it were a new female friend?
As flint92 already pointed out, "head" is a slang term used in Ireland, usually for a male acquaintance ("howya, head!"). Probably a shortened form of "head-the-ball", and equivalent to "skin" ("he's a dacent skin").
There is no obvious female equivalent, and it can't really be assumed that you would use "ceann" in the same way in Irish.
While many sentences on Duolingo are a little bit "odd", choosing an odd interpretation when there is a straightforward interpretation is not good learning practice.
The "ceann amháin nua" in this exercise refers to a thing, not a person.
That's unfortunate, because you can be almost certain that when you encounter the phrase ceann nua in Irish, it means "a new one", not "a new head", and therefor whoever decided to allow the "literal" translation was actually doing a disservice to learners.
I knew the correct meaning but couldn't resist. Surprised it was accepted.