Well, this would differ site to site. But its mostly by an evaluation of your record, activity and how you carry yourself in key situations and your ability to handle them. Its not something that should be determined by just one instance or a few, but derived from an evaluation that has been reoccuring since this member joined.
What have they done in this time?
Overall, there is no single attribute than an honest adminstrator should be looking for, but rather several. If it was just one attribute then many more would qualify than actually should and we'd have more people in authoritive positions than necessary.
I think its a combination of how the candidate has technically proven themselves, and what type of person they are. If they're unreliable, they're wasting space and abilities better suited to someone who can put them to better use.
And then of course theres the issue of what type of authority or position to give them.Theres different levels of that and different people that would be ideal for the varying positions.
A trend more and more common is the popular vote. Wherein people simply vote for who they want to lead them. This itself is a bag of mixed results and allows for too many loopholes and personal bias to get in the way. I don't recommend it.
In the end, it should be the adminstrator making this decision based only on what he/she wants and feels. Listening to what the people want is fine and including them is fine, but they're influenced by many elements. The call of the adminstrator is to be able to make these types of decisions without giving in to influence.
I've seen alot of site owners lose respect and legitimacy over this very issue, and its pathetic. If only they had listened to themselves before the whimsical demands of the lot. I've seen sites go under because the leaders basically allowed the members to run things, and make pretty much every decision that needed an opinion. Theirs were taken as absolute in place of the staff even, all in an effort to appease the members and keep them coming.
I think, in the end, you have to weigh what the site needs versus what it wants. A balance must be found, and that would be true for modship.