Female equivalent of 'son' to call kids

If the person speaking to the girl is not related, a common thing would be to say "young lady, you know that..."

If the speaker is her father, "young lady" may seem too formal. Her dad would probably just call her by her first name, as in "Jenny, you know that..."

This is, of course, assuming the girl is, in fact, a young lady. Usually, when someone refers to another male as "son", that male is a young man or boy. So, that is why I am saying that "young lady" would work for a female here.

Lass is defined by Meriam-Webster as:

a girl or young woman

This term could be substituted with son.

Lass, you know not to chew with your mouth open.

I think the closest and least condescending equivalent is Dear or My Dear. It is more casual than Young Lady, but perhaps less cliche, which was one of the required fields.

An affectionate or friendly form of address.

It adds an immediate connection, and has the same disarming effect that using Son in the way that you described does.

If the father is talking to his daughter, he would probably say "Honey, that is not how you treat your brother." If a man is speaking to a girl that is not related to him, I guess Cliff's answer above is correct, "Young lady, that is not how you treat your mother."