It's just that the normal expectation is she would have several friends.
We use the singular in contexts like He has no wife, or I have no car. We tend to use "do-support" or "got-support" anyway (He doesn't have a wife, I haven't got a car), so such assertions don't always sound completely natural regardless of the plurality.
Note that I have no son is likely to be bitter exaggeration from a father "disowning" his only son, whereas I have no sons is more likely if someone is lamenting the fact that he has no sons, whereas he would like to have several.