Add class to items in wp_list_categories()

Yes, it is possible. wp_list_categories accepts a (custom) walker that may handle this.

Put the following in your functions.php file:

class Custom_Walker_Category extends Walker_Category {

        function start_el( &$output, $category, $depth = 0, $args = array(), $id = 0 ) {
                extract($args);
                $cat_name = esc_attr( $category->name );
                $cat_name = apply_filters( "list_cats', $cat_name, $category );
                $link = '<a href="' . esc_url( get_term_link($category) ) . '" ';
                if ( $use_desc_for_title == 0 || empty($category->description) )
                        $link .= 'title="' . esc_attr( sprintf(__( 'View all posts filed under %s' ), $cat_name) ) . '"';
                else
                        $link .= 'title="' . esc_attr( strip_tags( apply_filters( 'category_description', $category->description, $category ) ) ) . '"';
                $link .= '>';
                $link .= $cat_name . '</a>';
                if ( !empty($feed_image) || !empty($feed) ) {
                        $link .= ' ';
                        if ( empty($feed_image) )
                                $link .= '(';
                        $link .= '<a href="' . esc_url( get_term_feed_link( $category->term_id, $category->taxonomy, $feed_type ) ) . '"';
                        if ( empty($feed) ) {
                                $alt=" alt="" . sprintf(__( 'Feed for all posts filed under %s' ), $cat_name ) . '"';
                        } else {
                                $title=" title="" . $feed . '"';
                                $alt=" alt="" . $feed . '"';
                                $name = $feed;
                                $link .= $title;
                        }
                        $link .= '>';
                        if ( empty($feed_image) )
                                $link .= $name;
                        else
                                $link .= "<img src="$feed_image"$alt$title" . ' />';
                        $link .= '</a>';
                        if ( empty($feed_image) )
                                $link .= ')';
                }
                if ( !empty($show_count) )
                        $link .= ' (' . intval($category->count) . ')';
                if ( 'list' == $args['style'] ) {
                        $output .= "\t<li";
                        $class="cat-item cat-item-" . $category->term_id;


                        // YOUR CUSTOM CLASS
                        if ($depth)
                            $class .= ' sub-'.sanitize_title_with_dashes($category->name);


                        if ( !empty($current_category) ) {
                                $_current_category = get_term( $current_category, $category->taxonomy );
                                if ( $category->term_id == $current_category )
                                        $class .=  ' current-cat';
                                elseif ( $category->term_id == $_current_category->parent )
                                        $class .=  ' current-cat-parent';
                        }
                        $output .=  ' class="' . $class . '"';
                        $output .= ">$link\n";
                } else {
                        $output .= "\t$link<br />\n";
                }
        } // function start_el

} // class Custom_Walker_Category

Then, change your code in category.php like so:

if (is_category()) {
    $this_category = get_category($cat);
    if ('' != get_category_children($this_category->cat_ID)) {
        echo '<ul>';
        $args = array(
            'orderby' => 'id',
            'show_count' => 0,
            'title_li' => '',
            'use_desc_for_title' => 1,
            'child_of' => $this_category->cat_ID,
            'walker' => new Custom_Walker_Category(),
        );
        wp_list_categories($args);
        echo '</ul>';
    }
}