This will add a new specialized Search preset link "View Unanswered Threads" to header.

Open template header_welcomeblock_member_search and add at the bottom:

<li><a href="{$mybb->settings['bburl']}/search.php?action=findunansweredthreads">View Unanswered Threads</a></li>

 
Now open your search.php file in an editor. Go to line no. around 1252, find this line:
elseif($mybb->input['action'] == "finduserthreads")

and change it to:
elseif($mybb->input['action'] == "finduserthreads" || $mybb->input['action'] == "findunansweredthreads")

Just after that, find this line:
$where_sql = "uid='".$mybb->get_input('uid', MyBB::INPUT_INT)."'";

and change it to:

 
    if($mybb->input['action'] == "finduserthreads")
    {
        $where_sql = "uid='".$mybb->get_input('uid', MyBB::INPUT_INT)."'";
    }
    else
    {
        $where_sql = "replies='0'";
    }

Now save the modified search.php and reupload to your forum root, replacing the existing one.

Result: