In the "read more" portion of this hint, you can read my original solution to this problem, left in place for posterity's sake. However, commenter bhas points out a much simpler solution in the comments to this hint: just Command-click on the Drawer icon, and the drawer will flip from one side of the window to the other, and stay on that side for subsequent launches.
This also works in TextWrangler.
Original solution:
After Tweeting about this annoyance, I found the solution courtesy of a Tweet from @methodphoto; he pointed me to this message in the BBEdit Talk archives, which discusses the MultiDocumentWindows:PreferredEdge preferences value.
With that knowledge, it became relatively simple to figure out how to put the drawer back on the left side of the window. Quit BBEdit, then run this command in Terminal:
defaults write com.barebones.bbedit MultiDocumentWindows:PreferredEdge -string Left
defaults delete com.barebones.bbedit MultiDocumentWindows:PreferredEdge

