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.
