Best How To :

It happens because the value of the selected option (and so of $_POST['billing_reason_for_purchase']) is actually the key of the array (in your example option_2) and not the text related. In fact this is the option tag created:

<option value="option_2">Academic</option>

You're saving only the key of your array.

So you need to retrieve the array of options even on the my_custom_checkout_field_display_admin_order_meta function to get the proper text.

You could copy the array in each function (but is not convenient to duplicate code), or put it in a global variable to be able to access it from anywhere (but then something else could change it), so instead you can use a function that returns the array:

function reasons_for_purchase () {
    return array(
       'option_1' => __('Personal', 'woocommerce'),
       'option_2' => __('Academic', 'woocommerce'),
       'option_3' => __('Small Business', 'woocommerce'),
       'option_4' => __('Large Organization', 'woocommerce')
   );
}

And then use it where you need it:

add_filter('woocommerce_checkout_fields','custom_override_checkout_fields');
function custom_override_checkout_fields($fields) {
   $fields['billing']['billing_reason_for_purchase'] = array(
       'label' => __('Reason for purchase', 'woocommerce'),
       'placeholder' => _x('Reason for purchase', 'placeholder', 'woocommerce'),
       'required' => false,
       'type' => 'select',
       'class' => array('form-row-first'),
       'options' => reasons_for_purchase()
   );
   return $fields;
}

add_action('woocommerce_admin_order_data_after_billing_address','my_custom_checkout_field_display_admin_order_meta',10,1);
function my_custom_checkout_field_display_admin_order_meta($order){
    $reasons = reasons_for_purchase();
    $reason  = get_post_meta($order->id, 'Reason for purchase', true);
    if( isset($reasons[$reason]) )
        echo '<p><strong>'.__('Reason for purchase').':</strong> '. $reasons[$reason] .'</p>';
}