Show/hide image in each list individually with same class
You need to find the .name item that is in the same block of HTML as the one being hovered.  One way to do that is to go up the parent chain from the one begin hovered to get the li tag and then use .find() from thereto find that .name item in that block.  You can use this code to do that:

$('.name').hide();

$(".first").hover(function() {
    $(this).closest("li").find(".name").stop(true).show();
}, function() {
    $(this).closest("li").find(".name").stop(true).hide();
});