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 onlyjust a person who studies abstract mathematics,. There are plenty of other occupations that involve the same skills and thereduties as pure mathematics. There is a lot of mathematics needed into other similar subjects, like computer science. AndMoreover, the interdisciplinarity of computer science should help her in getting such a position. Finally, as well as original thinking, and ability to formalize and being a hardworker are skills 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...