Let's start with the bar as high as possible:
Fundamental changes to the way I think
When I think about my favorite answers across the network, the one thing they all share is that the answer changed the way I think about a question. For instance, I asked about a Python design decision that struck me as odd and the answer gave me a sudden burst of insight into the philosophy behind the language. Or an answer on Kurt Gödel's Incompleteness Theorem that completely reframed my understanding of the already strange and wonderful idea. Or (more locally) an answer that took a step back and examined a passage in a refreshingly straightforward manner.
There's no objective standard for this, of course, and every reader will identify different answers if only because we start from different places. But it doesn't take many of these sorts of answers to make this whole endeavor worthwhile. Since not all answers can achieve this level of greatness, the next thing I look for is:
Answers that teach
Our highest rated answer simply explains, very patiently, why the Bible didn't carry out the digits of π. As you go down the list, you see answers that don't just provide an answer, but actually try to help readers learn something about the topic. Answers like these are perfect candidates for being paradigm-shifting for some people. If the top answer to a question isn't in this category, I'm left deeply unsatisfied.
I really think we do a good job of upvoting these sorts of answers. If an answer is accepted, it's likely to be of this type. But some questions just don't lend themselves to these types of answers. We really dig deep into the texts and often there is no prior knowledge to draw on. This long tail is really the great promise of this site. In order to answer obscure questions, we may need to settle for:
Answers that demonstrate learning
Thankfully, most of our answers at least fall in this category. People are just giving questions their best shots with the tools they happen to have acquired. For me, I've read a lot of history, Christian commentators, and some non-Christian interpretations of the Bible. Throw in a few online language tools, many years of small-group studies, and innumerable sermons and I can take a crack at a lot of questions. But since I don't know the answer before I start writing, I'm likely to make errors. In fact, screwing up is the sign that you are really learning and not just repeating what someone else has said.
If you pick a handful of answers at random, it's likely that all of them will be trying to learn, but aren't really authoritative. But the beautiful thing about Stack Exchange is that there are lots of ways for people to help each other construct great answers out of merely adequate ones. If I make a bad assumption about Hebrew grammar, I know there are people who will leave comments or even edit my answers to fix the problems. Sometimes answers that fail to teach (but show the learning process) will prompt other people, who know better, to write authoritative answers.
For me, this site is all about learning the Bible and the philosophy of interpretation. I'm good with all types of answers that at least make a good faith effort to move our understanding from darkness into the light.