So my wife freaked out this morning. She realized that the voicemails she had saved from her mom had been deleted off her her phone as we upgraded her from an iPhone 6 to a 6s. The good news is that I had been doing backups of her phone, locally on her computer through iTunes. I browsed this forum and found this post, with the positive reviews of this application:
http://www.macroplant.com/iexplorer/
I downloaded the free trial and discovered that I could easily browse through all of the iPhone backups I had made, even the encrypted ones (with the password to decrypt the backup, of course). To my wife's delight, every voice mail that was on her phone had been saved, all the way back to 2012! With the free trial she could listen to the messages, but not export them to files / iTunes. The $40 price tag to license the product was well worth it, as I was able to easily select the voice mail messages and save them to a folder on the Desktop. From there she could rename the messages to something meaningful, and then they could be imported into iTunes for long-term retention and easy playback.
Thanks for the suggestion. It really "saved the day" for both my wife and for me. :-)