What does "to give out" mean when talking about a person?

Well, I think taking "never gives out" to mean "never breaks down/never ceases to function/always keeps going" makes the most sense here.

I am not exactly sure if this is a valid explanation (as it is referenced from Wiktionary) but it does fit in well with the Meaning of the song.

To Give Out : (Ireland, idiomatic) To complain, sulk, chastise.

Based on the above, the line -

She never gives out - She never gives in - She just changes her mind

seems to all fit in with this meaning.

I'm Irish.

There are two contexts for this phrase giving out.

Give out to someone

To give out to a person means to speak angrily. For example, when the meal arrived it was cold so she gave out to the waitress. She was giving out to her husband because he always came home late from the pub.

Give out about something

In the Billy Joel song it means she never complains. If you are annoyed by something you might give out (complain) about it. For example, Pat was a complainer, he was always giving out about his mother and his job. Stop giving out about your job and do something about it.