You don't have enough of a load on the output of the SSR.
Most SSRs contain a snubber network that consists of a resistor and capacitor in series and then connected in parallel with the output terminals of the SSR. The capacitor allows some leakage current to flow.
This leakage current is enough to illuminate neon lamps and will even cause some 120 Vac strobe lights to flash periodically. Connecting a small incandescent lamp (5 Watt Christmas bulb) across your load terminals is usually enough to swamp out that leakage current.
Note: the snubber network is required for the triac inside the SSR to not false-trigger when certain conditions are just right. But the snubber does introduce its own set of problems.
According to the datasheet, the G3MB-202P is a discontinued zero-crossing switched AC SSR.
So, when it is commanded to turn on, it will delay by up to 1/2 of an AC cycle until the next zero crossing. So, at 50/60Hz it can delay by as much as 10msec or 8.33msec
When it is commanded to turn off, it cannot turn off until the current crosses zero. This can be as much as 10msec or 8.33 msec again, but not necessarily in phase with the voltage.
For a resistive load you if you commanded it on for a bit more than 10msec it would be guaranteed to turn on, and commanded it off for a bit more than 10msec it would be guaranteed to turn off. Using say 10.5 msec for each, that's 21msec or about 47Hz. In reality you would get beating between commands and response frequencies with wildly varying amounts of power, so perhaps 1/10 of that frequency or about 5 or 6Hz is more reasonable as a maximum, but even that is a bit high.
If you want to get 'even' and fairly-beat free power (for example for a PWM heater control), normally 0.5Hz is about right (gives you 200 half cycles per period at least).
Nothing you can do by commanding the relay will damage it with a simple load like lights, but you may get undesirable (or perhaps interesting) variations in light if you cycle it too fast. Some will probably appear as smooth pulsations in brightness, which might be quite okay. The life of the lights might be shortened a bit if they're incandescent, but they're only Christmas lights.
According to one review on DX:
Most likely (as no datasheet or part list to confirm), it uses a Triac or similar for the actual load control switching, which means that no current can be flowing for it to turn off. On AC, this is simple as the AC signal crosses the Zero point between Positive and Negative. On DC, the DC signal needs to be off for that to happen. It's a catch 22.
So no, you need a DC switching SSR. This one will not work well for you.