What is the theoretical lower mass limit for a gravitationally stable neutron star? I ask here intentionally not for the size of the smallest possible observed size of neutron stars, which corresponds approximately to the well-known Chandrasekhar-limit for the upper limit of the white dwarfs. This is defined by the minimal size of a stellar core to collapse into a neutron star, instead of white dwarf.
But, I think this is not the smallest possible neutron star mass - it is only the smallest mass that can be produced by stellar evolution processes.
For example, black holes have also a lower limit: the Tolman-Oppenheimer-Volkoff limit, which is around 1.5-3.0 Solar masses. Corresponding to that, the known smallest black hole is observed to be around 4 Solar masses. But this doesn't define the smallest possible size of a black hole, it only defines the smallest black hole size which can be formed. Theoretically, even Earth-sized or much more smaller black holes could exist, but there is no known process which could create them. Despite the extensive searches for micro black holes, nothing was found.
By analogy, I am asking is this a similar situation for neutron stars? What is the minimal mass of a neutron star, which could remain stable? Is this mass smaller than the Chandrasekhar-limit?
 A: We think that most neutron stars are produced in the cores of massive stars and result from the collapse of a core that is already at a mass of $\sim 1.1-1.2 M_{\odot}$ and so as a result there is a minimum observed mass for neutron stars of about $1.2M_{\odot}$ (see for example Ozel et al. 2012). Update - the smallest, precisely measured mass for a neutron star is now $1.174 \pm 0.004 M_{\odot}$ - Martinez et al. (2015).
The same paper also shows that there appears to be a gap between the maximum masses of neutron stars and the minimum mass of black holes.
You are correct that current thinking is that the lower limit on observed neutron star and black hole masses is as a result of the formation process rather than any physical limit (e.g. Belczynski et al. 2012 [thanks Kyle]).
Theoretically a stable neutron star could exist with a much lower mass, if one could work out a way of forming it (perhaps in a close binary neutron star where one component loses mass to the other prior to a merger?). If one just assumes that you could somehow evolve material at a gradually increasing density in some quasi-static way so that it reaches a nuclear statistical equilibrium at each point, then one can use the equation of state of such material to find the range of densities where $\partial M/\partial \rho$ is positive. This is a necessary (though not entirely sufficient) condition for stability and would be complicated by rotation, so let's ignore that.
The zero-temperature "Harrison-Wheeler" equation of state (ideal electron/neutron degeneracy pressure, plus nuclear statistical equilibrium) gives a minimum stable mass of 0.19$M_{\odot}$, a minimum central density of $2.5\times10^{16}$ kg/m$^3$ and a radius of 250 km. (Colpi et al. 1993). However, the same paper shows that this is dependent on the details of the adopted equation of state. The Baym-Pethick-Sutherland EOS gives them a minimum mass of 0.09$M_{\odot}$ and central density of $1.5\times10^{17}$ kg/m$^3$. Both of these calculations ignore General Relativity.
More modern calculations (incorporating GR, e.g.
Bordbar & Hayti 2006) get a minimum mass of 0.1$M_{\odot}$ and claim this is insensitive to the particular EOS. This is supported by Potekhin et al. (2013), who find $0.087 < M_{\rm min}/M_{\odot} < 0.093$ for EOSs with a range of "hardness". On the other hand Belvedere et al. (2014) find $M_{\rm min}=0.18M_{\odot}$ with an even harder EOS.
A paper by Burgio & Schulze (2010) shows that the corresponding minimum mass for hot material with trapped neutrinos in the centre of a supernova is more like 1$M_{\odot}$. So this is the key point - although low mass neutron stars could exist, it is impossible to produce them in the cores of supernovae.
Edit: I thought I'd add a brief qualitative reason why lower mass neutron stars can't exist. The root cause is that for a star supported by a polytropic equation of state $P \propto \rho^{\alpha}$, it is well known that the binding energy is only negative, $\partial M/\partial \rho>0$ and the star stable, if $\alpha>4/3$. This is modified a bit for GR - very roughly $\alpha > 4/3 + 2.25GM/Rc^2$. At densities of $\sim 10^{17}$ kg/m$^3$ the star can be supported by non-relativistic neutron degeneracy pressure with $\alpha \sim 5/3$. Lower mass neutron stars will have larger radii ($R \propto M^{-1/3}$), but if densities drop too low, then it is energetically favorable for protons and neutrons to combine into neutron-rich nuclei; removing free neutrons, reducing $\alpha$ and producing relativistic free electrons through beta-decay. Eventually the equation of state becomes dominated by the free electrons with $\alpha=4/3$, further softened by inverse beta-decay, and stability becomes impossible.
Edit2: There are claims in the literature that a minimum mass or even sub-minimum mass neutron star could be created by tidal stripping during the merger of two neutron stars (or maybe a black hole and a neutron star?) - e.g. Colpi et al. (1989); Yudin et al. (2023). These would then explode in some sort of weak supernova-type event, perhaps masquerading as a kilonova.
