“Order notes” will show on the invoice PDF, so the customer would be able to see these notes. However, what you could do when you need to add a note about an order is go to the customer’s page, click the ‘Record activity’ button, then chose ‘Note’, where you can then enter text and refer to the order in question. You can even backdate this note to make sure it appears next to the order in question in the dated list, if you’re adding the note a while after the order was processed.
If you’re looking to add a private note directly to a customer, there are a few things outlined in Luke’s response below. These are not per order, but might be useful to you anyway.
If none of these solutions work for you, you could put in a feature request for a private order note on our feature requests page.
Unfortunately, it’s not currently possible to export a list of orders that have order notes. If this would be really helpful, it might be possible to add the ‘Note’ field to the Raw Data Explorer report, where you could then export this list to excel and simply remove the orders that don’t have a note there.