Good news, that option is already available to everyone.
1) Click on the Profile button at the top of any forum page (such as this one).
2) On the left side, under "Modify Profile" menu, one option is: "Personal Message Options". Choose that.
3) On that page, there is a check box: "Save a copy of each Personal Message in my outbox by default." Check it.
4) Done.
If you only want to save a copy of specific messages going out, you can do that as well. Leave the option above unchecked.
Then, at the bottom of the "Send New Message" message screen, there is a check box that reads: "Save a copy in my outbox". Check that box before you send it.