I was called by a friend's widow on the weekend. Her house had no power due to a wide spread power outage. The generator was running, but there was no power to the backed up circuits.

I had her check the breaker in the in the generator enclosure. It was on. I had her turn it off, wait 5-seconds and turn it back on. Still no power. 

The System Set light was flashing. From the manual, this indicates loss of Utility power. Looks normal.

I had here remove the transfer panel cover. The transfer switch was in the utility position. That's as far as I went with troubleshooting over the phone.

When utility power came back on, the power to her house was restored, including all the backed up circuits.

I plan to go there this weekend and was going to simulate a power failure and do some tests. I was looking for suggestions beyond the following:

-Check for the generator to start, run and transfer. If no transfer:
-----Check the generator AC output for correct voltage/frequency
-----Check for the transfer signal for a on terminal 23
-----Check for +12 volts on terminal 194 (should be 12-volts between 25 and 194)
-----Check the small cube relay to see if it was "pulled in".
-----Check the coils on the transfer switch for continuity

I do have a copy of the Diagnostic and repair manual Part# 0G4338

Anything else that you might check while there?

Thanks in advance.

Manually start the generator and check for 240VAC between terminals E1 and E2 in the transfer switch with the generator running. (Transfer switch can be in either position, voltage should still be there.)

Trouble can be in the transfer control circuit, which it looks like you have a good handle on in terms of troubleshooting, or in the generator not delivering power. The control logic from the relay uses the power from the source to which the transfer will occur to energize the actual transfer solenoid. In other words, the transfer-to-generator coil gets its power from the generator and the transfer-to-utility coil gets its power from the utility. Each coil is in series with a microswitch that cuts power to the coil immediately when the transfer is complete. Those coils pack a lot of punch to make the transfer extremely fast but they are designed for VERY intermittent duty.

Thanks JayH for the ideas. 

I have the next version of generator that this lady has, but most of the same principals apply. I'm trying to get there this weekend and start doing some tests.
I have seen things as simple as a wire popped off of the coil cause this before. 
I second the response from grsthegreat, that is the most common problem if the breaker is good and the unit is producing voltage.
Just an update on this. I went to my friends place yesterday. It's been a couple of years since I checked the valves, so I adjusted those before starting it up (adjusted while engine is cold).

I then manually started the generator and measured the AC output voltage (unloaded). It was about 250 VAC line-line.

I then simulated a utility power fail by turning off the Utility AC power feed to the transfer switch. The generator started, then transferred as normal. I restored utility power and the transfer switch transferred back to utility and the generator shut down (with the typical delays).

No problem was observed. My friend did tell me that the day it was running but didn't transfer to generator power, that there were several short power failures and the generator started several times in a short time period. 

It appears to me that the generator got "out of sync" somehow that day.

My question now is:
If it "gets out of sync" in the future, what's the best way to get it to transfer to generator power? Should I instruct her to shut the generator off on the generator panel toggle switch, wait a minute and turn it back to Auto?

If I were there, I would probably try shutting off all the loads, perform a manual transfer with the yellow handle, then turn the load breakers back on (ensuring transfer is not done under load). I expect that once Utility power is restored that the switch will transfer back.

Note: This is the older, GEN 1 generator with just the LED's (no display).
I would shut the unit down just like clearing a fault then go back to auto.  It should go ahead and start if the power is out and transfer or if it has not transferred back to utility it should do so.  I have seen this a few time with multiple short outages where it seems to get confused.
Thanks for your reply Peddler. Your advice is much appreciated (and respected).
We have also seen this behaviour from Gen 1's during multiple short outages. That being said just yesterday I had a transfer relay "pull in" as it should during transfer, however the contacts in the cube made poor connection and transfer didn't occur. It really threw me for a loop until I finally just replaced it as it was my last real option. Problem solved... I had never seen a relay fail like that before.