**SOLUTION - Updated: 03/03/2015** - Thanks to /u/G.M.

The below adds a new field to the Term Edit Page and saves the value into the un-used ( for the moment anyway ) `term_group` field which can then be used to order terms. In the below hooks ( actions ) you'll need to replace `TAXONOMY_SLUG` with your actual taxonomy slug. 

    /**
     * Append Fields To Term Edit Page
     * @param Term Object $term
     * @param string $taxonomy
     */
    function term_order_field( $term, $taxonomy ) {
      ?>
    	<tr class="form-field">
    		<th scope="row" valign="top">
    			<label for="meta-order"><?php _e( 'Category Order' ); ?></label>
    		</th>
    		<td>
    			<input type="text" name="_term_order" size="3" style="width:5%;" value="<?php echo ( ! empty( $term->term_group ) ) ? $term->term_group : '0'; ?>" />
    			<span class="description"><?php _e( 'Categories are ordered Smallest to Largest' ); ?></span>
    		</td> 
    	</tr>
      <?php
    }
    add_action( 'TAXONOMY_SLUG_edit_form_fields', 'term_order_field', 10, 2 );
    
    /**
     * Save Term Order
     * @param int $term_id
     */
    function save_term_order( $term_id ) {
    	global $wpdb;
    	
        if( isset( $_POST['_term_order'] ) ) {
    		$wpdb->update( 
    			$wpdb->terms,
    			array(
    				'term_group' => $_POST['_term_order']
    			),
    			array( 
    				'term_id'	 => $term_id
    			)
    		);
        }
    } // END Function
    add_action( 'edited_TAXONOMY_SLUG', 'save_term_order' );

**Add Term Order cell to Term List**

    /**
     * Add Column To Show 'Term Order' Field
     * @param array $columns
     * @return array $columns
     */
    function term_order_header( $columns ) {
    	$columns['order'] = '<center>Order</center>';
    	return $columns;
    }
    add_filter( 'manage_edit-TAXONOMY_SLUG_columns', 'term_order_header' );
    
    /**
     * Give 'Term Order' Column A Value `term_group`
     * @param string $empty
     * @param string $col_name
     * @param int $term_id
     * @return string
     */
    function term_order_header_val( $empty = '', $col_name, $term_id ) {
    	if( isset( $_GET['taxonomy'] ) && 'order' == $col_name ) {
    		$term = get_term( $term_id, $_GET['taxonomy'] );
    		return "<center>{$term->term_group}</center>";
    	} else {
    		return '0';
    	}
    }
    add_filter( 'manage_TAXONOMY_SLUG_custom_column', 'term_order_header_val', 10, 3 );

**Hook Documentation**

 - [action `{$taxonomy}_edit_form_fields`](https://developer.wordpress.org/reference/hooks/taxonomy_edit_form_fields/)
 - [action `edited_{$taxonomy}`](https://developer.wordpress.org/reference/hooks/edited_taxonomy/)
 - [filter `manage_{$screen->id}_columns`](https://developer.wordpress.org/reference/hooks/manage_screen-id_columns/)
 - [filter `manage_{$this->screen->taxonomy}_custom_column`](https://developer.wordpress.org/reference/hooks/manage_this-screen-taxonomy_custom_column/)

---

Below is an out-dated solution that works better if you have multiple Term Meta Values you would like to save. If you only need to save Term Order, the above solution is best.

    /** Add New Field To Category **/
    function extra_category_fields( $tag ) {
        $t_id = $tag->term_id;
        $cat_meta = get_option( "category_$t_id" );
    ?>
    <tr class="form-field">
    	<th scope="row" valign="top"><label for="meta-order"><?php _e('Category Order'); ?></label></th>
    	<td>
    		<div id="catOrder">
    			<input type="text" name="cat_meta[order]" size="3" style="width:5%;" value="<?php echo (isset($cat_meta['order'])) ? $cat_meta['order'] : '0'; ?>" />
    		</div>
    		<span class="description"><?php _e('Categories are ordered Smallest to Largest'); ?></span>
        </td> 
    </tr>
    <?php
    }
    add_action('protax_edit_form_fields','extra_category_fields');  
    
    /** Save Category Meta **/
    function save_extra_category_fileds( $term_id ) {
    	global $wpdb;
        if ( isset( $_POST['cat_meta'] ) ) {
            $t_id = $term_id;
            $cat_meta = get_option( "category_$t_id");
            $cat_keys = array_keys($_POST['cat_meta']);
                foreach ($cat_keys as $key){
                if (isset($_POST['cat_meta'][$key])){
                    $cat_meta[$key] = $_POST['cat_meta'][$key];
                }
            }
    		update_option( "category_$t_id", $cat_meta );
    		$wpdb->update($wpdb->terms, array('term_group' => $cat_meta['order']), array('term_id'=>$term_id));
        }
    }
    add_action ( 'edited_protax', 'save_extra_category_fileds');

Calling it:

    $procats = get_categories(array('taxonomy' => 'protax', 'hide_empty' => 0, 'orderby' => 'term_group'));

**Edit** - You're also able to order them in the admin panel by using this filter:

    function change_term_order( $orderby, $args, $taxonomies ) {
        if ( is_admin() && 'protax' !== $taxonomies[0] )
            return $orderby;
    	
        $orderby = 'term_group';
        $args['order'] = 'ASC';
        return $orderby;
    }
    add_filter( 'get_terms_orderby', 'change_term_order', 10, 3 );

**Edit** — The function change_term_order should return $orderby, not the $args array