My reading of the datasheet is that the trigger current is the load current required to trip the zero crossing detector logic in order to turn on the load. I think it is separate from the input voltage required to activate the zero crossing detection circuitry.
In any case using a buffer like the ULN2003 to isolate the BBB GPIO pin from the SSR is a good idea.
Also, the current measured by your Kill-a-Watt is the 120 VAC input current draw by the power supply. It should produce 5 VDC at up to 2A on its output which is used to power the BBB.
I would connect the GPIO pin to the input of the ULN2003 channel you are using (and remember to connect the unused inputs and the ground pin (pin 8) to the BBB GND). When the GPIO pin goes high (3.3 V) the ULN2003 input will draw about 1 mA from the GPIO pin. In turn the ULN2003 will turn on and its output will go low (to about 1 V above GND). This output should be connected to pin 4 of the SSR (as shown in the Application Hints section of the datasheet). Pin 3 of the SSR should be connected to the 5 V supply pin on the BBB. The ULN2003 will draw current current through the SSR input and the SSR should turn on after the next zero crossing. When the GPIO pin goes low (0 V) the ULN2003 will turn off and the SSR should turn off after the next zero crossing. To unsure the SSR turns off when the ULN2003 turns off you should connect a 1K pull-up resistor between the SSR pins 3 and 4. This resistor will draw about 4 mA when the ULN2003 is on. Finally, to ensure the ULN2003 is not destroyed by any inductive flyback (unlikely here, but possible), you should connect the COM pin (pin 9) to the BBB 5 V supply.
This will provide about 4 V on the input to turn it on (when the GPIO pin is high), and 0 V on the input to turn it off (when the GPIO pin is low). These values are greater than the min ON voltage (>2.4 V) and less than the max off voltage (<1 V) specified for the SSR.
I hope that is clear, a simple schematic would be better.
HTH
Dennis Cote