Now, of course, there have been many blog posts, notably by local admins and wikia staff on what makes a good admin, and what criteria there are to chose a good one. However, this is of course in light of the usual encyclopedia wiki, which generally have topics with a limt; i. e. a wiki about a TV show can only have so many pages before building of the wiki is mostly done.
Now, under these wikis, an admin is kind of like the person running the place. Check for or respond to reported vandilism, go through the Admin Dashboard to check everything every once and awhile, make sure discussions are controled. Of course, an Admin is an integrated part of the community, but still, they more watch over it. Note this paragraph is a very dumbed down on what an admin at these places is, because that's not what this is about.
However, there are also wikis with or based around original content (notably fanon). Now, even if on "normal wikis" admins build content when new stuff comes along, but original content has many members creating a limitless ammount of pages. Here, being an Admin is different.
There are flows of new content coming in, and imporant, an administrator is generally (or should be) one of the people in the most discussions, controling and checking through new content, and writing much content. Infact, an Admin that just does their job but isn't a frequent of the main body of the community is actually probably doing their job badly. Mind you, an Admin not doing admin things is bad to. Don't give an admin powers just because they're a major part of the community, of course.
Chosing an admin should be choice made by all the current admins, and on bigger wikis agreed upon by the community, the biggest members of which are, again, already admins. A shortlist for potential new admins just between the current admins could work to. Questions to ask: Is this member in many discussions, if the wiki has many? Does this member contribute content a lot? Would this member understand and use admin powers? Is another Admin needed?
Just the two cents of an admin from a smaller one of these communities. Add opinions or points in the comments.