The current drawn by the SSR from the trigger circuit is significantly less than a mechanical relay. I'll try adding a resistor across the input terminals to generate some additional current from the ECU output.
Hopefully that will work.

Admin
I have a bag full of MUR415 diodes. Will wiring a few of them in parallel be good enough? They are rated at 4A forward current.

Yes, that will be fine.
 

Admin
The current drawn by the SSR from the trigger circuit is significantly less than a mechanical relay. I'll try adding a resistor across the input terminals to generate some additional current from the ECU output.
Hopefully that will work.

Measure the resistance of the old relay coil and add a resistor in line with the input for the SSR, that should allow cars circuit to believe a normal relay is in place and prevent the error light. I assume this is similar to the case of installing LED indicators without resistors and they behave in the failure mode.

Thanks mate. I'll give it a try later today.

Hey everyone. I wasn't able to get it to work with the SSR. No matter what I did, the PS ECU didn't like it and kept throwing an error.
For those of you that are interested, here are the measurements for the input side of the working Toyota relay - 

•  5.5V
• 680mA
• 7.7 Ohms

The SSR only draws 7mA so I thought that the ECU could be tricked by putting a 10 Ohm resistor in parallel with the SSR input. This drew 614mA and the SSR would switch OK and turn the pump on for a few seconds, though it would then turn off and the PS ECU would show an error. There is something that the PS ECU doesn't like though I can't figure out what it is.
I ended up putting in a high current 12V mechanical relay, and the 5.5V input seems to drive it OK and the PS ECU doesn't throw an error. 
Here are some photos of my finished product - 
 



 
 If anyone wants to buy a near new SSR, it is $30 ($50 new) and I will throw in a mounting bracket for free! 😊

Interesting read. 
 
Every time I go to start my SW20 the power steering pump runs for about 30 seconds.
Could this be the same relay at fault.
 
To be truthful I don't even know if the power steering is even working on the SW20.
I do know its it's much lighter to steer than my AW11.