Changing from name=xxx to old_name=xxx is the correct thing to do when a business moves out. It keeps the information about the building such as its street address and also preserves the mapping history. 

The old_name no longer appears on the map (when the tiles next update). When a new business opens there, you can add a new name=xxxx tag to the existing way. But it's still useful to keep the old_name tag so that history is not lost and the old_name can still be searched.