The basis would be getting a dean's approval (or lack of a denial). That is the level at which "Exception Handling" is normally done at a university (above professor, below vice president), and it would normally be the dean of the department that housed the graduate program (some grad programs span multiple departments, which makes it more complicated).

Likely, there are also rule systems that would need to be circumvented; for example, while the admissions program for a graduate program would accept the student, the registrar or other entity might then negate that acceptance due to standing policy. At that point, I wouldn't be surprised if it reached the vice president level of the university (VP of the School that housed the department).

Every university handles this slightly differently. Exceptions can often be made.

To qualify you would normally have to do something truly exceptional or show a pattern of continuously excellent (grad student level). From what I can tell (I am not a math expert) she is considered a 'prodigy' and the problem she solved would require considerable skill, in a way that predicts continued successful performance at the university professor. The history of prodigies who skipped various steps of the normal schooling path is mixed- personally, I think people undervalue the importance of spending time with people in your age group during the 15-25 range, but honestly, without knowing more detail, it's hard to say.