Does she really need to become PhD Student in Mathematics? Why doesn't she try with something different?
I mean, a mathematician is, in my experience, not only a person who studies abstract mathematics, and there is a lot of mathematics needed into other similar subjects, like computer science. And the interdisciplinarity of computer science, as well as original thinking, are really needed and rewarded in this field!
To say my experience, I graduated in Mathematics (in the field of arithmetic geometry) but now I'm a PhD Student in computer science... and despite my lack of knowledge in specific topics, like programming, I feel I'm really helpful here!
Moreover, in some areas of Mathematics the research borders are quite far, so starting "late" could imply facing with difficulties in the future... or at least, I feel that this is what the most mathematicians think about this, and probably is the reason why she's having hard times in finding a PhD position in Maths...