Update!
The last three attempts to connect to my Audible app have been successful every time. I primarily changed my iPhone prep and the order in which I connect to apps. I can't promise success, but I have received ZERO Connection Errors over the last couple days when loading up the Audible app.
My Setup: Hyundai Elantra using OEM cable + Lightning adapter cable connecting to an iPhone 5s with iOS 8.3
Prep: Before connecting the phone, I go into multitasking (double-Home button) and I close down all apps (upward swipe) that can potentially use audio. These include but are not limited to Apple Music, Audible, Amazon Music, Youtube, etc.
Connecting in the car:
--Start the car (OEM + Lightning cable already connected)
--Verify the stereo is set to XM or FM radio (both have worked)
--On the iPhone, get to the home screen and then connect the cable
--At this point, the phone connects and music starts playing from my main playlist. It always plays the first song alphabetically in the playlist. (This connection used to cause a Connection Error most of the time)
--With the music playing, open the Audible app and select an audiobook. The music fades out and the audiobook starts playing. Success! (Sometimes the playback would pause the first time. Toggling the play/pause button or using the 30 second reverse clears up the issue)
Overall my process hasn't changed much. The only thing that is different is closing down all of the audio-related apps (including Audible) before connecting the cable.
Hopefully this helps...good luck!
-R