Sorry for the delay! I hope this will still be useful.
Here's the design I've come up with. It seems to work OK in simulation. You may need to adjust it once you've built it up, to accommodate component variations. I will explain this below.
The circuit uses three transistors, and some resistors, capacitors and diodes. It is powered from the +12V source on the left; the top and bottom rails need to be connected to pins 6 and 7, respectively, of your 7-pin connector.
The circuit's input is on the node labelled Vin, at the left end of R1. This needs to be connected to pin 4 of your 7-pin connector.
At the right hand end, the inductor labelled Relay_coil represents the coil of the relay. This coil should be rated for 12V DC and should have a power requirement of 0.4W or less (equivalent to a resistance of 360 ohms or higher).
Here's a circuit description.
The input voltage passes through R1, and is clipped by D1 and D2. These components protect Q1 from noise, surges, or spikes on the signal. R2 pulls Q1's base to around 0V when nothing is connected to R1.
Q1 operates as an emitter follower; its emitter follows the input voltage, with a constant positive offset of around 0.7V.
At the base of Q2, this voltage is combined with feedback via R8 from the output. When Q2's base reaches about 0.65V, Q2 conducts and forward-biases Q3, which starts to conduct and pull its collector voltage upwards. This increasing voltage is coupled through R8 and reinforces the increasing voltage on Q2's base, i.e. it provides positive feedback, which causes the Q2/Q3 pair to quickly switch to fully conducting. Q3 energises the relay coil. In this state, Q1's emitter voltage must fall to a lower voltage before Q2 will turn OFF. This makes the circuit immune to noise around the decision threshold.
The graph shows the input voltage in green, and the voltage on Q2's base in blue. The input voltage is a triangle wave from 0V to 0.5V and back again, with a low-amplitude sinewave superimposed on it, to simulate noise.
The points marked A and B on the blue trace correspond to the turn-on and turn-off of the relay. At point A, when Q3 conducts and energises the relay, extra current through R8 causes Q2's base voltage to jump up sharply. At point B, when Q1's emitter voltage has dropped low enough for Q2 to turn OFF despite the extra current through R8, Q3 turns OFF and the current through R8 disappears, causing a sharp falling edge on Q2's base voltage, marked B.
The circuit activates the relay at an input voltage of around 380 mV, and deactivates it at an input voltage of around 220 mV.
The voltage thresholds are dependent on the characteristics of the transistors used, which vary somewhat due to manufacturing tolerances. So you may find that the voltage thresholds are not accurate when you've built your circuit. In this case, you can replace R4 and R5 with a 50k preset potentiometer aka trimpot, with its wiper to Q2's base, and the other ends to Q1's emitter and the 0V rail. This will allow you to adjust the threshold voltages up and down, but not the distance between the upper and lower trigger voltages.
You can calibrate the circuit using a potentiometer to supply an adjustable voltage to the input. Set the trimpot initially about half way. Monitor the input voltage while you adjust the potentiometer, and see what voltages the circuit switches ON and OFF at. Adjust the trimpot so that both thresholds are comfortably inside the 0V/0.5V range of the input signal. Probably ideally they should be centred around 0.25V. If they are too far apart, you can increase R8 (to 1.2 megohms or 1.5 megohms) to make the high and low voltage points closer together.
I hope this makes some sense. Let me know if you need more info
Again, sorry for the delay.
