I have the same issue in imessage on my macbook and the easiest "quick-fix" that I have found, is to delete the contact name that the message is to while still in the message screen. This clears the conversation since it no longer is "to" anyone, and then you can delete the (blank) message thread. If you then type in that contacts name in the search bar, the old thread will still be there, just not stuck to the top anymore.
I don't know if this will work on phones/tablets as well since I have never had the same problem there but I would expect that it should.