If you don't mind modifying the store module itself you can do the following (for Store v 2.5.1)
In file \system\expressionengine\third_party\store\src\Service\ShippingService.php line 139, in function get_order_shipping_methods
> foreach ($methods as $method)
> if ($rule) {
add line
$option->rule_description = $rule->name;
In file \system\expressionengine\third_party\store\src\Model\OrderShippingMethod.php line 53, in function toTagArray
add line
$attributes['rule_description'] = $this->rule_description;
In file system\expressionengine\third_party\store\src\Model\ShippingRule.php line 16 in protected $fillable = array(...
add 'name' string to array so line looks like:
'percent_rate', 'min_rate', 'max_rate', 'enabled', 'name');
Then add {rule_description}
to your {shipping_methods}
tag. Example:
{shipping_methods}
<li>
<input type="radio" name="shipping_method" class="shipping-methods" value="{method_id}" id="shipping-method-{method_id}" {if method_selected}checked{/if} />
<label for="shipping-method-{method_id}">{method_title} {method_price} <em>{rule_description}</em></label>
</li>
{/shipping_methods}