Will Gaia detect inactive neutron stars? Will the astrometric precision of the Gaia space telescope be able to detect the gravitational influence of cold old solitary neutron stars on the movements of stars? At least in a statistical sense to estimate how common they are around here, which could say something about how large the population III stars were.

And if an inactive neutron star could be precisely located thanks to its gravitational influence, would it be possible to observe it directly?
I will throw out a suggestion that could hardly be considered an "answer", but it would be interesting if someone could put flesh on the bones of it.

When I first read your question I thought you were asking whether gravitational perturbations from an unseen neutron star would cause a measurable effect on the positions of other stars. I think the answer to this is certainly no unless the neutron star was so close to another star that they were (or could be practically considered as) a binary system.

Binary systems featuring neutron stars could definitely exist (they won't be common though) and Gaia should see them if they are close enough and the companion star is bright enough. A variant of this would be a neutron star that just happens to drift (probably at high speed) close to another star, producing a notable change in the photocentre of the visible star over the 5-year Gaia mission. This is unlikely because close encounters between unrelated stars are very unlikely.

But then I started thinking about lensing. Even the presence of planets in our solar system causes a gravitational lensing signature - the positions of stars are distorted by $\sim 1-10$ microarcseconds close to the positions of planets and this is detectable by Gaia. The distortion can be spotted because of course the planets move with respect to the background stars.

In GR, the lensing angular deflection is
$$\alpha = \frac{4GM}{c^2r},$$
where $r$ is the closest approach of the light to the lensing object of mass $M$. For a lensing object at distance $d$ and a background star at an angular separation of $\theta$ from it, then
$$\alpha = 0.04 \left(\frac{M}{M_{\odot}}\right)\left(\frac{d}{\rm 1\ pc}\right)^{-1} \left(\frac{\theta}{\rm 1\ arcsec}\right)^{-1}\ \ \mu{\rm \ arcsec}$$

Neutron stars are of order 1-2$M_{\odot}$; the closest they might come is a few pc. The astrometric precision of Gaia will be about 20 microarcsec for a star with $V =15$, falling to about a milli-arcsec at $V\sim 20$.
The signature of a neutron star moving with high proper motion might be that it passes very close in the foreground to a bright star, causing its position to wobble backwards and then forwards by an amplitude given by the above formulae.

To get a 20 microarcsecond signature that might be detectable, a neutron star at one pc would have to pass within 0.003 arcseconds of a $V=15$ star.  The surface density of such stars varies according to Galactic latitude but is around 1000 per square degree for stars of $V\leq 15$ (Bahcall &amp; Soneira 1980) at intermediate Galactic latitudes. If the neutron star has a proper motion like Barnard's star of around 10 arcseconds/year, then it traverses 50 arcseconds during the Gaia mission. If background stars are evenly spaced, the chance of coming within 0.003 arcseconds of one is about $10^{-5}$ (if I've done my sums right).

So, almost no chance of seeing this.