"doesn't that have some potential to solve the storage problem which besets renewable leccy?"
Only with some breakthroughs that will make the discovery of the semiconductor small beer. That's because electricity to hydrogen back to electricity involves multiple conversions plus intermediate physical storage.
So state of the art at the moment is about 60-70% efficiency on large scale prototype electricity to gas plants - that's being done now, and is a hugely impressive achievement. Mainly that efficiency is a result of the dissociation of water to get the hydrogen. You'll only improve that if you can magically improve the dissociation technology, and I'd be surprised if we'll see major progress on that. A big part of the problem is that you only want the H, not the O, so the energy embodied in the dissociated oxygen is lost when it is vented. Technically you can capture and store the O, but the problem is that it's economic value is lower than the marginal cost of storing and distributing the oxygen to those who want it.
You then have problems of compression and decompression of hydrogen (uses perhaps 2-7% of embodied energy), the higher of those where you either have multiple compression/decompression cycles (eg distribution and transport use), or where you have to heat the compressed gas to decompress it (as you will in industrial scale plants).
If you can use the stored hydrogen in a grid-scale fuel cell, what energy efficiency might you hope for in practice? Let's plump for 50%, you can improve this by running in CHP mode, but that adds heat output, not more electricity, and requires a heat distribution network (the efficiency of large scale fuel cells is only marginally better than a modern gas turbine). All the talk of fuel cells as 90% efficient ignores the fact that they produce heat and electricity, and to be that efficient you need to be able to use both outputs in their entirety. In the case of heat that's very difficult in the real world.
So here's the rub: For every 1 kWh that goes into the power-to-gas plant, you get 0.3 kWh of electricity out of the fuel cell. So you need three times as much generating capacity upstream of the power-to-gas plant, and that's expensive; You need a power-to-gas plant, they don't come cheap; and you need in my example an expensive grid scale fuel cell (and an expensive heat distribution network if you want to raise the efficiency to a still middling 70%).
So the problem is that you need lots more capital intensive plant, and I can't see R&D materially bring the costs down by much. You *might* reduce the storage costs for hydrogen with nanotech. You will only improve the efficiency of the fuel cell if you come up with a miraculous recovery system for low grade heat (and nobody's done that economically despite a century of looking).
Technically storing (say) wind power as gas is easy - you can visit plants doing this today. What you can't do is magic away the problems of low grade losses in the various conversion stages, nor the need for multiple volumes of kit that cost huge amounts of money. If your dream is renewable power storage, then the problem becomes one of suitable sites for renewables.