Cool, thanks for the responses. I think the automation problem could be solved pretty well with some time and a learning algorithm. If you use all of the sensors available (iPhone's mic, computer mic, accelerometers, GPS, mouse/key activity, etc) and start with some untrained model at first, you can get feedback from the user each time a call is suggested. If the user often says no in some situation, the model updates. So the model is constantly computing the probability that the user is "free," and depending on the priority of a call and the free probability will be more or less aggressive about suggesting it. Model can be manually updated by user ("don't suggest calls when I'm at my girlfriend's or parents' house unless they are with people in group X"). I think this would lead to a lot of business calls happening late at night when everyone is up but doesn't think others are up.
As to the point about free time being useful for all sorts of subconscious processing, mental and otherwise: I do agree that there is a lot of subconscious processing that happens when showering, sleeping, washing the dishes, etc. The question this raises is: are you doing the same kind of processing while talking to someone? I can se the answer being no, but can also see it being yes.
I do think this would lead to a little more catching up with friends, but as long as you have cool friends with good ideas that can help you think this might also be a boost in your ability to achieve in the long term :)
All this said, there's still the argument that this is not the best use of developer time for making productivity apps. Other ideas?
Lastly, for this to be something that happens, I would need to pay someone else to develop it since I don't have the time. I actually think I know the perfect person. But the question is, do we think people would pay for the service? And also, how valuable would this be if only three people you knew had it? I guess it could still just suggest calling people when it knows you are free to see if they'll answer. But I think the value goes way down then, so that means there is a gimungus chicken-and-egg problem to solve, and I'm not sure how to do that.