Mining a neutron star -- what do you get? In science fiction you get neutronium which supposedly makes darn good armor--which of course has absolutely nothing to do with reality.
You hook up your Acme star spinner (pay no attention to the energy needed) and spin the star up so the surface gravity at the equator goes negative and start picking up the pieces.
At first you're going to get the normal matter that was on the outside, but once that's gone what happens?  With the pressure removed it's obviously going to change state, but into what?  Simple decay to hydrogen?  Given the density this doesn't strike me as the right answer.  Does it fuse to the lowest energy level, yielding iron and nearby elements?  Given the neutron density do we get a massive r-process capture, giving us a radioactive hell of row 7 elements?
Edit:  Based on what has been said so far I realize I'm not really describing it right.
The deepest parts of the surface matter obviously get quite a neutron flux and get pushed down the table and since there can't be solids at that point you will have mixing, it will push at least the denser stuff down the table at least to bismuth.  How far into the surface that neutron flux goes would say how much of it gets converted.
The issue comes down to how fast that conversion happens.  On a long enough time scale you get the sort of stuff that's thrown off by neutron stars going splat (but does that actually contain a bunch of stuff past bismuth that decays before we see it?), but what time scale is that?  As we pull off fresh stuff how far along the process does it get?
 A: As Quillo ProfRob and John Doty already pointed out, the general mining problem is that the dense matter in a neutron star will decay/transform as the pressure (or equivalently in this case: baryon chemical potential) drops.
Well, there is an exception for the stability argument at lower pressures, especially for neutron stars. The existence of Strange Stars is discussed in the field of nuclear astrophysics, which is a star containing (or purely being build of) matter containing strange quarks. It is also discussed that strange quark matter is absolutely stable, even at zero pressure (in media, this idea became known more widespread with the concept of "Strangelets"). Let's ignore for one second that both (existence of strange stars, stability of strange matter at zero pressure) assumptions are far from being settled facts, the neutron star mining idea is a promising way to obtain strange matter.
[I totally agree that this answer contains, although being seriously discussed in the field of nuclear astrophysics, more speculative aspects of compact stars. I felt that this added a useful information here because the question mentioned "science fiction" in the first paragraph.]
A: What happens to decompressed deeper material is that it decays into a soup of nuclei and neutrons, which then sort themselves out via the R-process. The result is neutron-rich radioactive heavy elements that beta decay into more stable elements over time.
A: This is not really mining, as the neutrons and protons and electrons in the inside of a neutron star recombine to give you nuclei when the pressure drops to zero. The solid crust of a neutron star contains (very heavy) neutron-rich nuclei arranged in lattice domains. At the interface between the core and the crust you have the so-called pasta phase (highly deformed nuclear clusters). All this stuff is comprised of neutrons and protons (the fraction between the two changes as you go deep, increasing the pressure). Charge neutrality is guaranteed by a sea of relativistic electrons, more or less like in a metal, but with a lattice of exotic heavy nuclei. Differently from the terrestrial solids, these neutron-rich nuclei are fully ionized.
The thumb rule is: as you compress, proton and electrons combine to give more and more neutrons. Finally, after the pasta you reach the core: this is a homogeneous fluid comprised of protons, neutrons and relativistic electrons (homogeneous cold catalyzed nuclear matter). This makes sense: the nuclear clusters are so compressed and close that they merge together in a homogeneous fluid. Some muon may appear (the exact density at which this happens is debated and depends on the still unknown details of the nuclear strong force). As you go deeper and deeper things are uncertain: you may find hyperons, a quark-gluon plasma, or other exotic stuff (we really do not know).
All this stuff exists because of the pressure and temperature conditions in a neutron star interior: remember that (differently from the atomic nucleus) a neutron star is held together by gravity, not by nuclear forces (nuclear forces create the pressure needed to sustain it against gravity). So, when you spin a neutron star up, you also change its density and pressure profile, changing also the composition (beta reactions take place).
In general, catalyzed nuclear matter has the tendency to recombine to give you the "usual" elements when you lower the pressure (or density): as @rob pointed out in the comments, this is well studied because of neutron star mergers (part of the matter is ejected and gives you metals like gold, platinum and many other elements.. this is called kilonova).
A: If you remove the pressure the material will just do its best to get to the state that has the overall lowest energy density.
Things that certainly will happen are that neutron-rich materials from the neutron star crust will just beta-decay back towards the low-density nuclear stability line. If those elements are super-heavy then they will be unstable to fission. Ultimately I would think you would get a mixture of heavy (heavier than iron), but stable elements.
This applies to the crustal material, but the point is that as you remove material from a neutron star, its mass diminishes and hence its radius increases. The surface gravity and pressure decrease and what was in the deep interior at high densities is now closer to the surface and at lower densities. This material will become crustal material if you allow that process to happen slowly enough that it can be done in pseudo-equilibrium conditions.
If you do it quickly then you are likely to get a fierce bonfire. That is because removing the pressure allows more protons and electrons to be produced from neutrons, but the protons are going to be able to start fusing with neutrons in electron-degenerate conditions. Whilst this is going to proceed very quickly and then carry on through subsequent fusion stages, there isn't enough energy there to unbind the star - so no catastrophic event, just a raging inferno on the surface.
A: The neutron star matter is a good model of what the universe looked like in the first few micoseconds after the Big Bang, just after the baryogenesis.
The neutrons in a neutron star are kept from decaying by the degenerate electron gas (both neutrons and electrons, as well as protons at some depth and below, are degenerate in a neutron star). Inserting an electron in this electron gas requires energy that the beta decay cannot supply.
If you release the pressure at quick enough rate, so neutrons are allowed to separate before some of them decay into protons and be able to fuse another neutrons, you will get exactly the Big Bang result - Hydrogen, some Helium, a minor amount of Lithium and even less of everything else.
If you are slow, you will get heavier and heavier nuclei along the "neutron drip line" as in the stellar r-process. You are free to use them hot or wait them to decay into more stable elements.
Needless to say, both strategies require a great deal of energy - you will in essence need to revert the whole stellar evolution (milions of years of star radiative output) as well as the supernova explosion that created the neutron star in the first place.
Happy cranking!
