I found a solution from another web site, where the author said that other connections sometimes interfere with Bluetooth device detection. Turning my Wifi Off was not enough, I had to get away from the Wifi - I literally hopped into my car and drove a block down the road. Then when I opened by Bluetooth setting and turned it off and on, my device name appeared in the "available devices" section. The I was able to easily pair with it.
Once the device is in the list, I don't have a problem any more - I can disconnect and reconnect easily