The Reversal badge is to give someone an incentive to answer a good question that got downvoted because others misunderstood it, or disagreed with the whole premise.
Discussion on SO about this:
I've had this happen to me before while trying for this badge both here and on Stack Overflow - just opening the page on startup doesn't seem to count as visiting the site - you actually need to navigate to another page before it will count your visit. Just viewing your profile or a random question once a day (or 10 if you'd prefer) seems to be enough.
Self accepted answers do not count, authoritative answer here and here, where it also states that only answers older than 10 days (with no votes) are considered. Also wiki answers and deleted answers are not considered.
See also: List of all badges with full description.