I've been wondering about the tenacious badge for some time, now I've been informed I got it. I understand the condition of winning the badge, but what is its purpose? What does it reward?
Tenacious - Zero score accepted answers: more than 5 and 20% of total.
It's kind of a consolation badge. It's recognition that you post useful, working, accepted answers even though at least 5 of these answers went without the upvotes they may have deserved.
An answer accepted with a score of 0 sounds to me like a working answer that didn't get the upvotes it was due. That is what this badge rewards.
Everything John says is correct, but in addition:
It rewards activity in low(er) traffic tags, where there tends to be a lower number of active users to upvote.
It also rewards those who are willing to help new(er) users with less than 15 reputation who do not yet have the right to upvote.