Assumptions:
* The nail is weightless
* It's relatively stiff but still elastic
* The hammer, or whatever you use to drive the nail is completely stiff, or at least much stiffer than the nail.
* The nail is strong enough to be driven through the concrete, i.e. if the force pushing the nail increases, the concrete will deform/break rather than the nail.
In order for the nail to move forward, it needs to be pushed with a sufficient force. If you were using a heavy hammer, this is how that would work: You accelerate the hammer towards the nail, i.e. store kinetic energy in the hammer head. That kinetic energy is then transferred to the nail, thus pushing the nail into the material. The material pushes back, so the nail will compress a little.
The easiest way to visualise that is by imagining the nail as a (fairly stiff) spring: The hammer hits one end, compresses the spring from the "head" end, and on the "foot" end of the spring, we can measure a force that rapidly increases while the hammer is being slowed down, peaks when the hammer has been stopped, and then reduces again while it bounces back.
To decide whether the nail progresses at all, we need to look at the peak force during the bounce. If this is below the threshold for the concrete, we're not getting anywhere. If it is above, then as soon as the threshold is reached, the "foot" end of the nail will start to move into the concrete, thus doing useful work. This means some of the kinetic enrgy from the hammer will be used to enlarge the whole rather than just bounce the hammer back. This also means that the theoretical peak force won't actually be reached because the concrete gives way before it gets there.
So, what decides how high that peak force is, i.e. whether we reach the threshold required to drive the nail forward? Going back to the spring analogy: The peak force depends on how far the spring is compressed, and that in turn depends on two factors.
Factor 1: The stiffness of the nail
Think of hitting a trampoline with a hammer, to drive it into the ground: The hammer slows down very gently while the nail compresses very far. if you plotted force over time, it would take a long time to stop the hammer, and the forces would be very low. With a very stiff nail (think of a thick, straight metal rod), there's almost no compression, and the hammer is stopped abruptly, so the peak force is huge. 621
This is why nails are usually straight metal rods, and why people prefer bouncing around on trampolines :)
Note: A critical look at the assumptions above shows that really we should be talking about the combined stiffness of the material/nail/hammer combination. Any elastic deformation makes reduces the peak load on the nail, as anyone knows who's tried to drive a nail into a thin, free-hanging plank of wood. -- I'll ignore this for now since the question was about nails in concrete.
Factor 2: The momentum of the hammer
Momentum is mass times speed. So if you double the mass of the hammer (while keeping the springiness of the nail constant), it will slow down the hammer at half the rate, meaning it will compress the nail twice as far, thus giving you twice the peak force. If you double the speed of a hammer, it will need twice as much slowing-down before it stops, so it would also compress the nail twice as much.
So, could a very light hammer work?
In very general terms: The peak force is proportional to nail stiffness times hammer mass times hammer speed. So to find out if the nail can be driven, you could start with the weakest hammer blow that would still drive the nail just a tiny little bit, and in proportion to how much lighter it gets, you have to either move it accordinly faster or stiffen the nail proportionally. In practical terms, steel nails don't get much stiffer unless you make them thicker, but that would increase the driving force they require. Maybe a different material could help, but there are no huge gains to be made, even with very expensive materials. This leaves the "hammer" speed. For every halving of the mass, you'd have to double the speed. This means "dropping" small objects wouldn't work unless you drop them very far, and their speed is not limited by aerodynamic drag. Hitting the nail with fast-moving small objects could work, of course.
In slightly more realistic terms
"hitting it with small, fast-moving objects" ... if you've ever used youtube, you'll have noticed slowmo videos of stuff being hit by bullets. 621