I've been asked to keep an order open. What does this mean?

When someone asks you to "Keep an order open" this means to simply not pack and send the items yet. Buyers like this option when they have some projects going with several wanted lists, they can order things without having to rush. 
You will want to keep the order status the same until they ask you to close the order. The buyer can add items to their cart and checkout multiple times like normal and it will keep adding "batches" to the order rather than marking them as separate orders. These batches do not need to meet the minimum order amount as they are adding to the original order. 
Also, you can invoice as many times as you would like and the buyer can still keep adding to the order. Once the buyer decides they are ready, you can invoice the final amount and change the status to "Packed" which will finalize the order and remove the option to add any more batches.
One other option that I have seen sellers use is to add "Keep my order open" as a custom shipping method in their store. You can set this as a custom shipping method here
We also recommend keeping the communication going between yourself and the buyer in the next few days to see when they are ready to close the order. An open flow of communication is always the best way to avoid a problem order.