Why was M87 targeted for the Event Horizon Telescope instead of Sagittarius A*? The first image of a black hole has been released today, April 10th, 2019. The team targeted the black hole at the center of the M87 galaxy.
Why didn't the team target Sagittarius A* at the center of our own galaxy? Intuitively, it would seem to be a better target as it is closer to us.
 A: Of course they targeted Sgr A* as well.
I think though that this is a more difficult target to get good images of.
The black hole is about 1500 times less massive than in M87, but is about 2000 times closer. So the angular scale of the event horizons should be similar. However Sgr A* is a fairly dormant black hole and may not be illuminated so well, and there is more scattering material between us and it than in M87.
A bigger problem may be variability timescales$^{\dagger}$. The black hole in M87 is light days across, so images can be combined across several days of observing. Sgr A* is light minutes across, so rapid variability could be a problem.
The penultimate paragraph of the initial Event Horizon Telescope paper says:

Another primary EHT source, Sgr A*, has a precisely measured mass three orders of magnitude smaller than that of M87*, with dynamical timescales of minutes instead of days. Observing the shadow of Sgr A* will require accounting for this variability and mitigation of scattering effects caused by the interstellar medium

$\dagger$ The accretion flow into a black hole is turbulent and variable. However, the shortest timescale upon which significant changes can take place across the source is the timescale for light (the fastest possible means of communication) to travel across or around it. Because the material close to the black hole is moving relativistically, we do expect things to vary on these kinds of timescales. The photon sphere of a black hole is approximately $6GM/c^2$ across, meaning a shortest timescale of variability is about $6GM/c^3$. In more obvious units:
$$ \tau \sim 30 \left(\frac{M}{10^6 M_{\odot}}\right)\ \ {\rm seconds}.$$
i.e. We might expect variability in the image on timescales of 30 seconds multiplied by the black hole mass in units of millions of solar masses. This is 2 minutes for Sgr A* and a much longer 2.25 days for the M87 black hole.
EDIT:  12th May. And here it is, an image reconstruction, published by the Event Horizon Telescope consortium (see here) for the black hole at the centre of the Milky Way. The image is a time-averaged composite reconstructed using a novel dynamical imaging process for about 10 hours of VLBI data.

