Relay module without pinout diagram

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.

Are you setting the pin on your microcontroller as an output? The behavior of the LED makes me think you did not. You might get a volt or so, enough to light the LED a little but not enough activate the relay. I had this problem recently and nearly tore my hair out, so I thought I should suggest this in case it is indeed your problem.