"No Balls"

Can anyone shed any light on the phrase "

Your understanding is correct--"a lack of courage".
Just a guess but I think the origin is quite sexist.
In the US women are discriminated against in jobs, pay and status. They are second class citizens to males.
The phrase is a reference to a woman's lack of a scrotum and, therefore, her lack of status since she does not have the ability to produce testosterone, that manly of all manly body chemicals.

We can fix this. How about -- "You don't have the ovaries to do that." Or, if someone is cowardly, call him or her "t*tless." What do you think?

I think the metaphor arose from a comparison between men and little boys, not between men and women, and is based on external anatomy, not on endocrine physiology. This belief is totally intuitive. I have no evidence to back it up.

I agree with R.Berg. It's rarely used towards woman. It is usually used towards men to demasculate them.