Maybe? I believe LH2 has about 35 mols of hydrogen per liter, while water is 55 mols per liter. Storing hydrogen as water seems practical from that perspective, but what of the power needed to split that water? I think you'd need quite a lot of power to split that much water fast (starting a few days before running the engine.) Splitting it slowly over time using solar energy would seem to still leave you with a storage problem, but perhaps a more tractable one.
Maybe instead of electrolysis, they could use heat from the reactor? Thermolysis needs 2500 C though.
NTRs are basically open-cycle gas cooled reactors. The thermal limit on the reactor temp is when does stuff start to melt. Project Rho[0] suggests that's the reactor temp anyways. But you need to be able to separate out the oxygen from the thermolysis stream, rather than just feeding the entire thing into your engine, both because your Isp would go to crap if you tossed the oxygen out too, and you'd have oxidizing your reactor problems. Though, you could just store it all as ammonia, and you get more hydrogen for your buck, and can probably just feed that all through the reactor.
Right. In a NTR the nuclear fuel has to be hotter than the hydrogen (or ammonia or methane or whatever) propellant so that the heat energy from the first conducts to the second. In a combustion energy the fuel and the propellant are the same substance so you try to limit conduction and can end up with propellant much hotter than the engine.
Wouldn't you just size the engines small enough that they instantly burn off the H2 as you crack it? The solar power should be even and constant so you can size the system to match. It is going to require a very large solar array, especially since your spacecraft is going to be really heavy with all of that water.
You could design a pulse detonation engine for this. Electrolyze water continuously; detonate it in a pipe every now and then. It's a very simple design that gives you quite a bit of performance for hopping in the asteroid belt. Specific impulse similar to a hydrolox engine, or slightly worse than regular hydrolox engines if operating stoichiometrically, although the detonation mode could compensate for that (detonation rocket engines can potentially get ~10% better Isp performance than "classical" rocket engines). However, you get triple the propellant density (water has ~1000 kg/m³; hydrolox is at around 340 kg/m³). This makes it much better compared to a classical hydrolox vehicle wherever gravity is near zero so that you don't need lift-off thrust.
Maybe instead of electrolysis, they could use heat from the reactor? Thermolysis needs 2500 C though.