A note An essay Ramblings from one of your current moderators
Moderators have significant latitude in their duties and abilities, ranging from closing questions to suspending users. But at the very heart of our (and I do realize I have changed person here, thank you very much) actions is the flag queue, in which we deal with posts and comments that other users have flagged. It is then, in my opinion, necessary for a moderator to properly understand what flags are for, and what response is appropriate.
In some cases, where a regular 3k+ user might vote to close, for example, or where a 10k+ user might vote to delete, a moderator may wish to refrain from action if there is no community or SE policy about the post in question. I should note that this is my opinion, and to be fair, there is also compelling reason to advance the opposing opinion that a moderator should act even in borderline cases since moderators are merely elevated users, and should not have their personal opinions about what belongs censored. Naturally, whichever side you wish to take is yours, but I suggest that what a particular candidate thinks about what a moderator should do with borderline cases be in your thoughts when you vote.
But to return to flagging, a good candidate will probably have, in their (and yes, I do realize I have used the singular they) time here, flagged a good many posts and had these flags marked helpful. A user who has historically refrained from flagging (a rough guideline is that below ten flags is too few) is a user who has historically taken little interest in site moderation, and while such a position is perfectly fine for users who only wish to contribute knowledge and have little interest in such things, it is a tenuous position for a potential moderator to take.
Besides a good historic flagging record, what should we look for in a potential moderator? Certainly, temperament is key—a candidate who is prone to name-calling or arguing is not perhaps our first choice as a moderator. Whether justified or not, such behavior can only provoke site users more, since this behavior is coming from a site moderator. Meta participation is important as well, but not crucial. A potential moderator should have taken an interest in guiding the site along the right path, but this can be expressed either on meta or by closing and deletion questions on the main site. And naturally, you should look to their nominations to see what their take on moderation is.
I also want to mention one thing that you may not necessarily want to place too great a weight on—reputation. Certainly, site activity is important, but reputation is more indicative of how much knowledge a particular user has contributed to the community than that user's previous contributions to site moderation, as neither close votes, deletion votes, nor flags have any effect on reputation. Nonetheless, passing the thresholds for close-vote privileges and delete-vote privileges and then using said privileges is something you may wish to take into account.
Naturally, everything I have mentioned here stems from my personal experience, even if I do not personally fit that characteristic (no one is perfect, after all), and does not represent the opinion of the moderators collectively. Any grammatical mistakes that I did not attach a parenthetical comment to were unintentional.