I'm thinking about light solar drones, and started wondering if you could hover while not using any power by flying downwind. Something similar to how prey birds hover while tracking the ground. It seems they can just keep their altitude without using energy, at least in strong headwind. Could this be a feasible way to "rest" the drone while charging the solar panels instead of having to land?
... wondering if you could hover while not using any power by flying downwind.
If you're anchored to the ground like a kite, then yes. Otherwise not of course, the relevant airspeed for a flying object or animal is the surrounding air, not the ground speed. When flying, upwind is wherever the nose is pointing.
Those beautiful birds of prey hanging still in the air are catching updrafts, like gliders do as well which seek them out - caused by winds being deflected upwards by dunes for instance, or by differently coloured surface areas which heat up the local air more, causing it to rise faster.
So for more info on where to "rest" the drone while charging, best talk to the local glider club.
You can 'hover for free' in a helicopter, but it requires the air surrounding the helicopter to be moving upwards, at the same rate that the helicopter is descending in 'autorotation'. Being in autorotation requires no engine power, and if the air surrounding the helicopter is moving upwards as fast as the helicopter is moving downwards - then technically the helicopter is not moving, and could be considered as 'hovering for free'.
I have experienced this twice during my helicopter flying career:
Firstly within a thunderstorm. There are horrendous updraghts and downdraughts within cumulonimbus clouds. I once flew a helicopter into one of these, where at times I was still climbing whilst in full autorotation using no engine power. Technically 'hovering for free'. However, this was shortly followed be a period of still descending whilst at maximum engine power! This is uncomfortable, not recommended, and I do not want to do this again.
Secondly, I have experienced hovering next to a mountainside, also whilst in full autorotation. This was near to the top of a mountain ridgeline, with very strong winds blowing up the side of the mountain, whilst rescuing a fallen climber. Again, the strength of the wind resulted in sufficient air blowing up through the rotors, that the result was the helicopter was in full autorotation, with no engine power. But again, this was uncomfortable, turbulent, not recommended, and I do not want to do this again.
As Koyovis' answer correctly states, finding updrafts (especially thermally generated ones or ones generated by orographic lift) is the only way you'll get an ability to remain in the air without expending energy unless you're tethered to something holding you in a spot relative to the ground. Of course, you can also remain aloft more or less indefinitely if you go into orbit, but good luck getting there.
As for using solar power, landing to charge is going to be your best bet anyway because the most efficient way to have a solar-powered drone - by far - is to have your entire solar power collection setup mounted on the ground and just use that to charge a battery on your drone. Solar power is not anywhere near the energy density that many people seem to assume it is. Using solar power for flight is only even remotely feasible if the vehicle is rather large, but also quite light and not trying to achieve high speeds because it needs a very large area to mount enough photovoltaic cells to get enough power. Solar power received at Earth is only about 1,000 Watts per square meter (about 1,000 Watts per 10 square feet.) However, that's only near noon at a latitude where the sun is directly overhead and with no clouds obstructing the sunlight.
For example, the Solar Impulse 2 managed to circumnavigate the world (with lots of stops) using solar power. However, the thing had a wingspan larger than a Boeing 747-8 and a total wing area of 270 m^2 (2,900 sqft,) but weighed only 5,100 pounds, being able to carry only one person with the remainder of the weight being batteries, solar panel cells, its electric motors, and its very weak structure for its size (designed to keep structural weight to an absolute minimum value without falling apart.) It took off at 19.5 knots (22.4 mph, 36 km/h) and it cruised at only 49 knots (56 mph, 90 km/h) in the daytime and a third less than that at night to save power. On its circumnavigation trip, it began on March 9, 2015 in Dubai and returned to Dubai on July 26, spending 23 days, 6 hours airborne and the rest of that time on the ground resting and waiting for favorable weather to continue. (Weather constraints are much tighter when your aircraft only flies 49 knots and its structure is designed to be just barely strong enough to not disintegrate in good-to-moderate weather.)
Aside from a few special-purpose use cases (for example, research or surveillance aircraft loitering at high altitudes at low or mid latitudes for extended durations,) on-board solar isn't really a feasible power source for flight, least of all for rotary-wing aircraft, which are far less energy efficient than fixed-wing ones.
If you have an airplane floating above a conveyer belt while the air is stationary, relativity tells us that that is equivalent to an airplane floating in moving air above stationary ground. In the first case, the conveyer belt clearly doesn't help the airplane stay afloat, so the fact that, in the second case, the air and airplane are moving at a different velocity than the ground also doesn't help the airplane stay afloat.
A vehicle can't get energy, relative to some homogenous medium, by interacting just with the medium (it can gain kinetic energy by either spending potential energy or fuel, but it can't gain net energy). There has to be some gradient for it to gain energy. A vehicle can gain energy relative to the ground by interacting with the air (that is, if it starts at rest with respect to the ground, and is pushed by the air, its energy relative to the ground can increase) if the air is moving relative to the ground, but to get energy relative to the air, it has to interact with some other medium such as the ground (a solar powered vehicle is interacting with sunlight, which for purposes of this answer is a "medium"). A vehicle can gain energy relative to both by interacting with both (for instance, a sail boat interacts with the wind through its sails, and with the water through its keel).