Jonathan, in Chrome you cannot select the text of a chat archive. I have had the same problem.
My workaround is this:
Open the chat archive (either from the campaign page or from inside the campaign, there is no difference).
Right click on the archive.
Click: save as
Select: webpage complete
Go to the file you saved
Right click on the file you saved
Select: open with
select: internet explorer
Now you may copy/paste it normally into a word document.
Drohems method also works, but if you want to copy and paste you will still need to open the downloaded file in internet explorer.
- Gauss