Giving wp_list_categories the class of the category

I would use get_categories() instead, so that you can have better control over the output of your list. And if I recall correctly, wp_list_categories calls get_categories anyway. You can use the same $args array in either function, and you should get the same categories as a result.

Then you can simply build your own unordered list like so:

// build category collection
$categories = get_categories($args);
$menu = '<ul>';

// iterate through your categories
foreach($categories as $category) {

    // you can also use $category->slug
    $menu .= '<li class="' . $category->name . '">' . $category->name . '</li>';

}

$menu .= '</ul>';
echo $menu;

Of course, you may build out as much markup to your <ul> and <li> tags as you wish (additional classes, anchors, etc).

You may even wish to mimic all the other classes that wp_list_categories adds (just so that you stay consistent). And if you use this in multiple places, it might even be good to create your own function.

For a complete reference of which properties are available in your $category object see the Codex.

Hope that helps!