While trying to troubleshoot why some kitchen electrical outlets in this
junction box weren't working, I did a bit of research. Apparently, there is
supposed to be a ground connection (other than the screw going through the
frame of the outlet to the box) from the GFCI outlet to the junction box.
(see photo:
http://webpages.charter.net/mrfixite...nction_box.jpg ) Can I
just run some wire from the green screw terminal on the GFCI outlet to the
box? Where would I attach it to the box? By the way, the problem with the
outlet was that the breaker tripped in the GFCI. There are three pairs of
wires in the photo. One is the incoming AC. One feeds another GFCI outlet
down the line. The last pair goes to a switch in the same junction box which
turns on some lights in the kitchen.

Well, let's see.

Having a ground-wire connection to the GFCI itself is a good idea, if
a good ground is available. It's not (I believe) strictly required -
a GFCI can be used on a two-wire circuit.

In your case, it does appear that the outlet box is probably grounded;
I see what appear to be bare ground wires entering at the back. These
wires *should* be securely bonded to the junction box - this would
probably be done right at the back, where the screws and wire clamps
are located. I can't tell if those ground wires are in fact securely
bonded to the box, or whether they're just sitting there.

If you're going to ground the GFCI, you'd run a wire from the green
screw terminal, to a connection point with one of those ground
wires... e.g. if the existing bare ground wire is screwed to the box
at some point, you'd want to connect the ground wire to that screw or
to another box-mounted screw not too far from it.

Now, the lack of a ground wire from the GFCI shouldn't tend to cause
the GFCI to "trip" mistakenly. A proper GFCI operates by measuring
"current imbalance" between the hot and neutral wires. If more flows
through one than through the other, then it's almost certain that some
current is flowing to ground (somewhere) rather than back through
neutral... and this is a Bad Thing (shock hazard, or shock occurring).

A GFCI's protection does *not* depend on measuring current flowing
through its own ground wire. That's why they can be used to protect
two-wire circuits, where there isn't even a good ground connection
available... this makes them useful for retrofitting older house
circuits for improved safety.

Having a good ground on a GFCI circuit is actually likely to make the
GFCI trip sooner, rather than later, if an electrical fault exists.
For example, if you have an appliance with a grounded case, and a leak
or short develops between "hot" and the case, the GFCI will trip
instantly. If the case isn't grounded through the third prong
(e.g. if there is no ground prong, or if it's cut off or defeated with
a "cheater", or if it's plugged into a 3-wire outlet with no actual
ground) then the fault won't result in current flow to ground... until
somebody touches the case and also touches something grounded. At
that point they'll get a shock, and (knock on wood) the GFCI will trip
fast enough to protect them from anything more than a scare.

So, if your GFCI is tripping, then I don't expect that grounding the
GFCI more securely is likely to change this. You may actually have a
current-leak-to-ground on one of the circuits you are protecting. Or,
possibly, interference from a strong local radio transmitter is
"falsing" the GFCI - it does happen.