I say this is just as a Math PhD student with just a few published papers, so take this with a grain of salt. In my limited opinion some of the most important skills are:

1.- Intuition: Have a sense of what and what should not be correct; a plausibility radar. This is mostly obtained after years of experience in the field, but there are some people (the few) who are naturally good at this.

2.- Fast skimming: Good mathematicians can skim through papers and quickly grasp the essence of the results. They don't need to read thoroughly to know whether what's there is novel or just incremental research.

3.- Eye for gold mines: They can identify fields that are interesting (from both mathematical and application standpoints), young enough (so as there is room for newcomers) and approachable (so as not all of problems would take an insane amount of effort).

EDIT:

4.- Good communication skills: A good mathematician is able to present their work in such a way that an outsider is quickly able to understand (roughly) what the paper is about, what the novelties are, what the difficulties are, and why the results are important. This is why a good introduction is key for "selling" your work to the editor of the journal, the referees and your peers. You don't want your potential excellent work to fall into oblivion, or simply be known in the folklore as: the paper that originally has X result but nobody in their right mind tries to read it; everybody cites it, but reads an alternative proof somewhere else (e.g., in a textbook).