  The standard LambdaCDM model based on General Relativity (GR) including cold
dark matter (CDM) is very successful at fitting cosmological observations, but
recent non-detections of candidate dark matter (DM) particles mean that various
modified-gravity theories remain of significant interest. The latter generally
involve modifications to GR below a critical acceleration scale $\sim 10^{-10}
\, m \, s^{-2}$. Wide-binary (WB) star systems with separations $> 5 \, kAU$
provide an interesting test for modified gravity, due to being in or near the
low-acceleration regime and presumably containing negligible DM. Here, we
explore the prospects for new observations pending from the GAIA spacecraft to
provide tests of GR against MOND or TeVes-like theories in a regime only
partially explored to date. In particular, we find that a histogram of (3D)
binary relative velocities against circular velocity predicted from the (2D)
projected separations predicts a rather sharp feature in this distribution for
standard gravity, with an 80th (90th) percentile value close to 1.025 (1.14)
with rather weak dependence on the eccentricity distribution. However,
MOND/TeVeS theories produce a shifted distribution, with a significant increase
in these upper percentiles. In MOND-like theories {\em without} an external
field effect, there are large shifts of order unity. With the external field
effect included, the shifts are considerably reduced to $\sim 0.04 - 0.08$, but
are still potentially detectable statistically given reasonably large samples
and good control of contaminants. In principle, followup of GAIA-selected wide
binaries with ground-based radial velocities accurate to < 0.03 km/s should be
able to produce an interesting new constraint on modified-gravity theories.
