Code for WordPress that adds a widget to select categories to be linked.

tps-widget-select-categories-to-link.php

<?php
/*
 * Add to WordPress a widget to select the categories to be displayed in a sidebar.
 * The links are ordered by the number of posts of each category.
 * Author: Tatiane Pires
 * Author URI: http://tatianepires.com.br/
 * Version: 1.0
 */

/*
 * This program is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 2 of the License, or
 * (at your option) any later version.
 * 
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 * GNU General Public License for more details.
 */

/*
 * Widget class
 */
class tps_categories_list_widget extends WP_Widget {

	function __construct() {
		parent::__construct(
			'tps_categories_list_widget',
			'Links de Categorias',
			array( 'description' => 'Exibe uma lista com os links das categorias selecionadas' ) 
		);
	}
	
	// Creating widget front-end
	// This is where the action happens
	public function widget( $args, $instance ) {
		$title = apply_filters( 'widget_title', $instance['title'] );
		$categoriesList = maybe_unserialize($instance['categories']);
		// before and after widget arguments are defined by themes
		echo $args['before_widget'];
		if ( ! empty( $title ) ) echo $args['before_title'] . $title . $args['after_title'];
		
		tps_categories_list_html($categoriesList);
		
		echo $args['after_widget'];
	}
			
	// Widget Backend 
	public function form( $instance ) {
		if ( isset( $instance['title'] ) ) { $title = $instance['title']; }
		else { $title = __( 'New title' ); }
		$selectedCat = maybe_unserialize( $instance['categories'] );
		// All categories
		$catArgs = array(
			'type'                     => 'post',
			'child_of'                 => 0,
			'orderby'                  => 'name',
			'order'                    => 'ASC',
			'hide_empty'               => 1,
			'hierarchical'             => 1,
			'taxonomy'                 => 'category',
			'pad_counts'               => false 
		);
		$categories = get_categories($catArgs);
		// Widget admin form
		?>
		<p>
			<label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Title:' ); ?></label> 
			<input class="widefat" id="<?php echo $this->get_field_id( 'title' ); ?>" name="<?php echo $this->get_field_name( 'title' ); ?>" type="text" value="<?php echo esc_attr( $title ); ?>" />
		</p>
		<p>
			<label><?php _e( 'Categories:' ); ?></label>
			<?php
			echo '<ul class="cat-checklist category-checklist">';
			
			foreach($categories as $category) {
				if( is_array($selectedCat) ) 
				if( in_array($category->term_id, $selectedCat) ) { $checked = 'checked="true"'; } else { $checked = ''; }
				echo '<li class="selectit">';
					echo '<input type="checkbox" name="'.$this->get_field_name( 'categories' ).'[]" id="cat-'.$category->term_id.'" value="'.$category->term_id.'" '.$checked.' />';
					echo '<label for="cat-'.$category->term_id.'">'.$category->name.'</label>';
				echo '</li>';
			}
			echo '</ul>';
			?>
		</p>
	<?php 
	} // function form
	
	// Update widget: replace old instance
	public function update( $new_instance, $old_instance ) {
		return $new_instance;
	} // function update
} // class tps_categories_list_widget

/*
 * Register and load the widget
 */
function tps_categories_list_load_widget() {
	register_widget( 'tps_categories_list_widget' );
}
add_action( 'widgets_init', 'tps_categories_list_load_widget' );

/*
 * HTML for widget front-end
 */
function tps_categories_list_html($categoriesList) {
	
	if( is_array($categoriesList) ) {
		
		// order by the number of posts in each category
		$categoriesListString = implode(', ', $categoriesList);
		$catArgs = array(
			'orderby'            => 'count',
			'order'              => 'DESC',
			'style'              => 'list',
			'show_count'         => 0,
			'hide_empty'         => 1,
			'use_desc_for_title' => 1,
			'child_of'           => 0,
			'include'            => $categoriesListString,
			'hierarchical'       => 0,
			'title_li'           => ''
    );
		echo '<ul>';
    wp_list_categories( $catArgs );
		echo '</ul>';
	} else {
		return;
	}
} // tps_categories_list_html
?>

Save tps-widget-select-categories-to-link.php on your template folder and add the following to the file functions.php of your theme:
require_once( get_template_directory() . '/tps-widget-select-categories-to-link.php' );

Preview of the widget in WordPress' admin.

Style the widget output with Font Awesome icons.
/* Categories */ .cat-item a:before { content: "\f07b"; font-family: FontAwesome; padding-right: 5px; } .cat-item a:hover:before { content: "\f07c"; padding-right: 3px; }