Considering we do not carbon copy anymore, should the abbreviation for copying additional persons on a letter be "copy" or should it still be "cc"?
Not necessarily. It's possible that "cc" doesn't actually mean "carbon copy" at all, but may come from the Latin for "with copy".
Since "cc" is still frequently used, particularly with e-mail, some people think of it as meaning "courtesy copy".
What do you mean by 'should'? In a sense it would be more accurate, but back in the '70s and '80s many of the IETF documents - IETF Home Page - that specified how e-mail could work (start looking here RFC Editor - home if you're feeling strong) used the prefix CC_ to refer to fields that concern the sending of copies; if your mail agent lets you do it, you can look at some of these internal fields, using some kind of 'Options' setting.) So the parts of the mail system that users see talk about 'CC', 'send a CC', 'Blind CC', and so on.
So although you might prefer not to refer to obsolete technology (carbon copies), the language that has evolved to cover e-mail forces you to. This isn't a new issue; it happened years before e-mail, with telephones; people were saying 'on the hook' long after the last candle-stick telephone (Google Image Result for http://www.corp.att.com/history/images/milestone_1919.jpg) was used - in fact, you still 'hang up' at the end of a telephone call. And the computer protocols that underly telephone systems - even in systems where the thing at the end of the line is not a telephone (say, a fax machine) - use expressions like 'an off-hook condition'. It may not be pretty, but that's language for you.
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Let's not forget just how many words that describe old technology are still in use today to describe new technology, changing their meanings.
Websites have pages; your computer probably has a desktop and a recycle bin; you have files arranged in folders; you move a cursor around the screen; you click on buttons; if you are a programmer, you have to know about line feeds and carriage returns; you open and close windows and choose commands from menus.