We're talking about this "unbanked" and where I've worked and seen, a simple mobile phone with GPRS (not smartphone, and not even 3g) has worked wonders. Perhaps if we can get Bitcoin to that stage, [...]
Indeed an interesting question, independently of the current Bitcoin usage pattern as "store of value" (it could be useful even for this purpose).
Of course the challenge is to build an app which is light enough to fit on such a "feature phone" but on the other hand doesn't sacrifice the decentralization and censorship resistance. Programming a M-PESA style Bitcoin app with a centralized provider is trivial.
I've worked with Bitcoin libraries in Python which have very lightweight components for the wallet storage, more so if you port them to C++ or another lower level programming language. The problem is perhaps the cryptographic stuff, e.g. the process of generating a new key pair and associated address, because for this process you need entropy and I don't know if a GPRS phone could generate that on a high enough level in a short time. However, you could perhaps let such a process run for a prolonged time, the user could do that while they're recharging the phone.
IMO technically it should be possible. I've searched for "most lightweight bitcoin wallet" but didn't get really a good answer.
I have to say it, maybe I said it too often, but the simplicity of M-pesa amazed me almost 20 years ago when I first saw. It still did 10 years ago, when I was STILL using a Nokia, while my peers were using even cheaper $5 second-hand Chinese rip phones. That you could pay with a free sms to buy a small packet of single-use shampoo on the streets or a cigarette (yes, a single cigarette) each costing less than $0.01... without any understanding necessary of how it really worked? That's how you serve the unbanked. (Although by that time some places charged a small fee and I understand these fees have become higher)
So to my technologically unconscious mind, and forgive my inaccuracies but it would appear that if you took all that work and asked the phone only to do the minimal required... perhaps a mesh network where all phones shared some of that computing power when on charging? I know it's really dumb to always think back on mesh but I saw how it worked wonders in areas with a lot of devices but very low capacity for range and bandwidth, could the same not be done in this way?
But yes, even an mpesa style Bitcoin app is trivial, but the side question remains, why go for BTC when you just stick with your pesas for almost free or sometimes free for instant payment right?