What signal to noise ratio is required for astrometry is a frequently asked question. It is best answered by considering the two steps of the astrometric process separately.
For plate solving, there are different requirements for the minimum allowable SNR of the reference stars depending on the specific methods you are using.
Case 1: If you are plate solving an image taken with a telescope that provides more or less perfect tangent-plane projection, then the best thing to do is solve with only high-signal stars. At 701 and 933 (stations with narrow fields but relatively deep magnitude reach), I'll typically limit the stars used in the solution to those stronger than around 5 sigma. If there are many such stars, I'll increase this value in an attempt to limit the number of stars used to about 20 or so. If there are not enough stars with sufficient signal, I'll reduce the signal requirements to four or perhaps three.
Under this regime, the goal is to get about twenty stars that are evenly distributed across the image and which are high-signal to be used in the plate solution.
Case 2: If I am plate solving an image taken with a Newtonian fitted with a coma corrector, or with an SCT fitted with a focal reducer, or with any other telescope that creates significant amounts of distortion, I will perform a solution with much lower signal stars. Typically anything 3 sigma or stronger can be used in the solution.
The reason is that I use a reduction software package which includes empirical terms for distortion modeling. The more data points that go into the model, the better the model works. The trade off is that the stars used in the solution are not as precisely fitted as if you had used only high-signal stars. But what you gain is a map of the distortion in your image that allows you to make much better measurements of objects at the center and at the edges of your field. The end result is much better astrometry than if I just used high-signal stars and assumed tangent-plane projection.
The amount of signal needed in your target for good measurements is entirely dependent upon what your target is.
If you want to make a useful measurement of an object that already has a good orbit (or a stationary object whose position is already roughly known), then you should expose so that the target has a lot of signal. I'd go with a minimum of five sigma and certainly wouldn't consider more than ten or twenty sigma to be too much. Of course, for high-precision astrometry, you need to worry about such things as using a good reference star catalog, avoiding gradients in the image, optimizing your image scale to your local FWHM, and so forth. In this case, a high-signal target is only one of several requirements for doing high-precision astrometry.
However, if you are measuring positions of an object that hasn't been observed in a while and consequently has an ephemeris uncertainty of many minutes of arc, measurements are going to be useful even if they are made barely above the noise floor. In such cases two sigma or even less is not unreasonable, as long as you can still centroid.
Lets say that in one of your images of a known target, an unknown asteroid appears. This asteroid is a 'lost' object that has a current ephemeris uncertainty of thirty minutes of arc (though you don't know this yet). You measure it even though it is only 2 sigma above the noise, and send the positions in. Your measurement is pretty poor - it has a 1 arcsecond residual - but its ok, because you've reduced the uncertainty at the epoch of your observation by a factor of 1,800. Congratulations!
In emergencies, such as recoveries of PHA's, just getting the position of the pixel that the object appears on can be 'good enough' to get observers at other stations to reliably target the asteroid; in such cases the signal doesn't even have to be good enough to centroid to generate a useful measurement.
On the other hand, there is no reason to provide low-signal measurements of well-observed objects. So the answer to "what signal-to-noise ratio is needed" really is "it depends."
The centroiding error can be estimated by using the equation:
Sigma = FWHM / SNR
where Sigma is the 1-sigma uncertainty of the position, FWHM is the full-width, half-maximum of the object's image, and SNR is the speak signal-to-noise of the object's image. The units of Sigma and FWHM are the same.