I guess it depends on how you define "cheap" - I found Hue to be markedly cheaper than many other places in Vietnam when I was there (2 years ago).
I stayed at Canh Tien Guesthouse, on Le Loi alley on the south bank. A large double room with ensuite was $10 US. The alley is lined with inexpensive guesthouses, so if you're looking for even cheaper, you should have no problem finding a shared bathroom room for less than $10. It's quite touristy round there, but if offers a host of inexpensive options, so you can just show up and take your pic.
Have you checked out the hotel listings here on Travelfish? http://www.travelfish.org/location/vietnam/central_vietnam/thua_thien_hue/hue Some of the reviews are a bit old, but they'll give you other options, including other areas of Hue.