Simply, it is by far the best of all possible aircraft fuels.

Its one major disadvantage for this, as for most uses, is its low mass density: it needs more room than diesel to store enough to be useful. The tanks need to be bigger, and either strong enough to contain high pressure, or well-enough insulated to keep it liquid.

Current aircraft store fuel in wing tanks not roomy enough for hydrogen. The most practical hydrogen-fueled aircraft would probably have a shape more like a lifting body than a submarine with skinny wings sticking out. So for best efficiency, we might need new airframes, but just using hydrogen yields major improvement. To oversimplify, the plane doesn't need to lift so much weight of fuel to cruising altitude, or carry it halfway across the world.

Tankage has improved radically in recent years with the development of aerogels, which make practical carrying liquid hydrogen that need not held be at high pressure, so can be in tanks that conform to an aerodynamically-practical shape. (Other sophisticated storage methods increase weight, so are more practical on the ground.)

Hydrogen is tricky to store for long or in large amounts, so the best way to use it is to produce it on demand where it is needed. A major airport would be a good place to produce it, as it could be piped directly into aircraft and used immediately. All the airport would need is water and power. Power can be collected by wind and solar over a wide area, delivered by transmission lines. The amount of water needed is negligible.

Producing hydrogen would also yield plenty of oxygen, which might just be vented; but by carrying liquid oxygen, aircraft could fly higher, faster, and more efficiently, or at least get to cruising altitude more quickly. As electric power continues to get cheaper, uses for it such as liquifying oxygen along with hydrogen get more attractive.

The first use will be in long-haul craft operating from a few major airports. It is possible that current large aircraft -- 747s, 777s, A380s -- could be converted, by using some of what is now cargo space for tankage, and replacing fuel pipes and pumps and, quite possibly, engines. It would be a big job, made attractive mainly by the extreme cost of qualifying new airframe designs.

The lost cargo space would be made up easily by the much larger weight capacity, as tens of tons less weight of fuel is needed. Cargo aircraft today often fly half-empty so as not to exceed their takeoff weight limit.

How is it better than generating biogas from biomaterial, something we already do at industrial scale? What’s the advantage of biohydrogen over biomethane?

Venting methane is worse, but that would be wasteful, too.

It’s not really about that, we don’t lack water or methane. The issue is rather about the energy required to produce given amount of hydrogen. Electrolysis requires 2-3 as much energy as steam methane reforming already if you just look at energy of chemical reactions, and moreover, steam reforming requires just heat, while electrolysis requires electricity, which again is more expensive than heat: we produce most of our electricity from heat, and the conversion efficiency from heat to electricity is around 20-30%, so it makes more sense to use all the heat directly instead of going through electricity production stage.

2. Huge efficiency gains are to be had, in electrolysis, via catalytic action, needing only transition from lab to production funded by sufficient demand for the hydrogen.

3. There has been great success with direct solar -> H2, needing, again, only sufficient demand for the hydrogen produced to be made industrially practical.

While duty cycle of solar is under 50%, when there is not sunshine there is very often wind. So 50% idle capacity is unlikely.

Catalytic conversion of electricity to hydrogen will probably exceed 90% efficiency in short order, and from H2 back to electric via fuel cell is already over 90%.

Liquified and pumped into aircraft, nothing comes close to H2 for efficiency. Likewise, in rockets.

When we finally get sensible carbon taxation, 6x cheaper than CH4 will be easy to achieve.

That’s because the alternative process to electrolysis of producing hydrogen, the steam reforming, uses heat, not electricity. I keep talking about heat, because electricity from solar or wind is competing not with electricity from natural gas, but with heat from natural gas. Given typical conversion efficiency of 20-30%, this means that the heat from natural gas is at the very least 3-5 times cheaper than electricity from natural gas, and in fact it’s even cheaper than that, because you don’t need expensive turbines to generate heat from natural gas, reducing your capital and maintenance costs even more. EIA doesnt expect costs of electricity from solar to go below electricity from natural gas by 2040 at all, much less 6 times.

When we finally get sensible carbon taxation, 6x cheaper than CH4 will be easy to achieve.

Yes, if you put a finger on a scale, you can achieve any result that you want.