It's hard to say exactly without seeing the full context but "to give her a way out" would mean something like someone is in a difficult situation or dilemma, and giving her a way out is providing her with the means to free herself from that situation, solve her problem, etc.

This doesn't always refer to a physical exit. In fact, they way it is used being "gave her a way out" makes me think it is not talking about a physical exit. In any tough situation, one from which a person wants release, a way out can be offered. This way out can be any alternative that allows someone to escape from obligation, or allows them to get away from the tough situation. It would be helpful to know the context, but whatever it is, look at the context where you got that. Most likely "she" owed a debt, had to face consequences for something, or had some very unpleasant task to perform. "He" probably offered an alternative so she would not have to pay her debt/face the consequences/perform the unpleasant task, etc. The alternative he offered her was the way out. By offering this alternative, he gave her a way out.

I'm not sure about this expression, but I think it means 'to help someone', 'to give a hand or to be handy', 'to save someone from a bad situation giving him or her a hand'. ---> To Show her the way out. I didn't know about this expression neither. I searched about it and I found this interesting video: http://www.youtube.com/watch?v=-GMnp54xMGE

give her a way out : means that someone gave her / opened /permitted her a way to escape or release her . example : he blocked the way out http://www.audioenglish.net/dictionary/way_out.htm