What is the general basis for skipping college and getting admission directly in a Ph.D. program? What does one have to generally do to qualify?

AFAIK there's no general procedure; it's just that the admissions committee can admit anyone whom they're convinced can do Ph.D.-level work, and that almost always involves getting an undergraduate degree first but in extremely exceptional cases it might not.

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.

For context, Stephen Wolfram dropped out of Eton to start at Oxford, then was admitted to Caltech's physics PhD before finishing his Oxford BA, receiving his doctorate at the age of 21.

(He was publishing papers in high school.)

I knew someone who was admitted to a Masters program at Stanford sans undergraduate degree. His employer at the time was a large donor.

Essentially, you have to convince a math professor that you able to complete the Ph.D program. There are a variety of ways to do that. Solving a major open conjuncture is clearly one of them but just getting to know one or another professors can work (I went to school with the semi-famous Paul Lockhart who also did this).

be actually gifted

this is what gifted looks like, not legions of students who are told they are "gifted" but are just pretty smart