Hi folks,
This issue completely disappeared for me after going into settings and manually turning Siri on. This can be deceptive, because I had never turned it off and in fact thought I had been using Siri since the iOS upgrade, so it never occurred to me to try this potential fix. But when I finally decided to check, Siri was indeed showing as "Off" in my settings. When I turned it on, the overall look and feel of Siri changed a little bit, and the black screen problem never returned. So this whole bug could be related to the way in which iOS9 upgraded Siri.
Try it, perhaps in combination with a soft reset afterwards. (iPhone 6 64 gb)