I think you could do something like this by adding a new column to the table that records details of the products for the clients. If you add a new column for archived you can set it to 1 or 0 to indicate that the product has been archived.
Once you do this you will then add this column to the query on the page and filter it for only products that have not been archived.
Then next part of this where you click on something to archive it will be like a regular update server behavior. Clicking on the button will trigger the update server behavior and you will just update the archived column to be 1.
The other part of this is to make sure that your query that shows the archived items selects the archived column and filters where it is set to 1. Please post back with any questions that you have on any part of this.