wireless doorbell schematics
I think the amplitude is greater than 0.5V, it is probably the speaker loading down the output to that level.
Remove the speaker and then send the output through a diode and capacitor to form a half-wave rectifier. This will convert the pulses into a DC voltage. Use this DC level to drive a transistor which then turns on the relay. This will give a momentary relay closure to approximately the same length of time they hold the button down.
If you want to have a latching relay that remembers its state, you need to include a flip-flop.