I find it interesting that the discussion has been about manufacturers dumping a bunch of R&D money into developing some sort of brand new ultra-cheap car. I guess some have, but that doesn't make a whole lot of sense to me due to the massive investment to overcome (although it overcomes the issue of updated safety requirements).
What I would think, especially for the USA, is you'd take the rental fleet (alternatively called the developing nation export) approach. Keep an old model on the market for as long as possible, and cut as much from it as you can. This could likely (in my head anyway) even be feasible with a larger car than would be possible with a brand new design, overcoming the issue Americans have with small cars.
Basically my vision for a super-cheap car is a Chevrolet "Classic."
If that were the case, I'd probably go with it. I'd expect such a vehicle to be better built and engineered than something originally designed to be bargain-basement.
Otherwise, if we're talking 'built from the ground-up to be cheap,' I'm not sure.