If there is a name for this, it must be somewhat obscure; I've not run across any such thing in my time on various metas. It's just referred to as unaccepting one and accepting the other, or changing where the checkmark is, or whatever seems idiomatic.
There is no badge for this. The closest is Populist, given when your answer has score of 23+ and more than double that of the accepted answer with a score of 11+, but that's really more the reverse, since if your answer is accepted before you get the badge, you're no longer eligible on that answer.
Basically, it's behavior that's already rewarded well enough by the rep system, for the most part, so badges do not seem necessary. (They might also encourage badgering* the asker for a change, which is definitely a very bad thing to have happen.)
*No pun intended. Originally.