Increasing the current to a solid-state relay?
If you plug three strings of lights into each relay you'll have 120mA (assuming the lights actually draw 40mA each). 

A 2K ohm 10W power resistor in parallel would also work. 

It's also possible the lights will work just fine as is, at least for a hobby type application. The consequences of them not work (maybe the lights flickering instead of lighting steadily. or some lighting of the LEDs when the relay is 'off' due to the built-in snubber) are minor, so it may be worth a try.