Combining multiple Illustrator files into one file

Theres a number of ways you could try this.  Depending on what the print shop needs of course.

If you have multiple artboards in a .ai file you could use the "save as" function and in the dialog box you'll see an option that says "save each artboard to a separate file".  Then you could select the files that result choosing only the ones you want to print again and (provided you're on mac) use acrobat's "combine" function to press all those files into one PDF.   This I believe would be the most efficient method.

** The resulting files would pair the original files name say.. "master" with the artboard name "1".  You can rename the artboards ahead of time with a script or manually to make sure your file output is organized and easy to navigate.

OR if you really want to have a combined Illustrator file of all the art but naturally it doesn't want to copy the artboards as you were expecting.... You could make place holder objects that stay on your artboards and are the same size as the artboard(Visible or Invisible). When you select all on that artboard it would select that item as well then when you paste in the new file select the placeholder object and use Object>Artboards>Convert to Artboard.

** If you find this is something you may become accustomed to doing you could also use the attributes panel on your invisible/visable object to input a note on that object that perhaps says something like "artboard".  You only have to do that once then can make it a symbol for reuse later or copy and paste that object as often as needed. If you make it a symbol you have to break the instance of the symbol before it remembers there's a note on the object. Then you can create an action that will select all objects with "artboard" as the note and it will then convert those to artboards all at once instead of manually doing it all the time.

          Action: Select Object "artboard", Convert to Artboards.

Just two possible humble solutions to consider.

Copy-paste in a new AI document with multiples artboards.

How about bringing the needed Illustrator files into InDesign? With the page tool in inDesign you can make your InDesign pages the same size as the needed Illustrator files. Then you can import only the artboards you need into the InDesign file. When you're ready to send the job out you can either do a collect within InDesign and send out only what is needed to print or even export a single pdf to send to the printer.