Lucy,
Don't know if you are still having this problem but for you and/or anyone else having the same problem, I think I found the solution.
I had two folders with the same name so to avoid confusion, I wanted to delete one of them and couldn't see how to do it, just like you.
I asked the question online and got no joy so I went to the AOL help pages. I eventually found a question on how to delete unwanted mail folders and it told me the following:
Go to the left hand pane and next to the Folders box, click on settings Icon. A Pop-up box appears listing all the folders and giving you the option to re-name or delete one at a time.
When I tried to do this, the settings icon was just not there and I was very frustrated. Then for some reason I don't quite understand, I clicked on the line on the edge of the left had pane and dragged it over to the right to make it bigger/wider. I then tried clicking on the folder tab again and there, as if by magic, was the settings icon. From there on it was as the AOL Help pages had said and I was able to delete my unwanted folder.
So that's the answer, stretch that pane and your pain (forgive the pun) will be at an end.
Hope this helps.
Alison