Here's an idea that could be called "I don't want to bother you, but I didn't want to forget to tell you" voicemail message feature. Okay, so I'm not great at coming up with names.
I know I've ran into a few instances where I'd like to leave a quick voice message for someone, without having to ring their phone and get into a conversation. Kind of a backend voicemail contact. They aren't bothered by the ring, and yet you are able to communicate your message.
Many might be asking, well "why don't you just text them?" Believe it or not, there are those of us that don't text all that often and find it cumbersome to communicate that way. It would take us 10 minutes to text the message, versus 30 seconds in leaving a voicemail.
This feature has been talked about by others for a few years now. Doing a quick google search will reveal how unprofound this idea is, but will also reveal some clues to the market potential.
One overarching problem I could see in having this type of feature on your phone. SPAMMERS. I can only imagine if your number gets put onto the spammers hotline. You know how that part of the story goes.