Hi, sorry for the delay. Here's a simple suggestion.
The voltage on C1 represents the state of the LOW/DRL input. Whenever LOW/DRL is low (0V), C1 is discharged quickly through R1 and D1. Whenever LOW/DRL is high, C1 charges relatively slowly through R2. So while LOW/DRL is low, or has the 56 Hz square wave on it, C1 is kept mostly discharged.
Only when LOW/DRL is steadily high will C1 charge up significantly. This corresponds to the "low beam" state of the signal on the LOW/DRL (which is one of the inputs to the headlight assembly). In the other relevant state (daylight running lights or DRL), the LOW/DRL signal carries a 56 Hz rectangular wave alternating between 0V and automotive supply, with an average voltage around 8.2V, which keeps C1 nearly fully discharged.
The time constant of R2 and C1 is 0.1 seconds which means (with a few approximations) that the C1 voltage will reach 63% of the applied voltage (automotive V+) after 100 ms from the last instant when LOW/CRL is low (0V).
Therefore, in DRL mode, C1 will be kept mostly discharged, and in the LOW mode (LOW/DRL constantly high), C1 will reach around 8.7V in 100 ms from the last low state on MODE/DRL.
The voltage on C1 is fed through D2 to Q1 which operates as an emitter follower. Its emitter follows the voltage on C1 with a two-junction voltage drop (around 1.4V).
Current through Q1's collector-emitter path forces C2 to follow the emitter in an upwards direction, but when the C1 voltage is low, C2 holds its voltage and discharges slowly through VR1 and R3. This provides the hold-on delay, which forces the relay to stay energised for a short time after the end of the LOW condition.
VR1 and R3 in series provide the R part of the time constant for C2. VR1 and R3 also form a voltage divider with its output feeding Q2 gate. Q2 has a gate threshold of a few volts (not a very well controlled parameter in MOSFET manufacture) and will conduct and energise the relay while its gate is above this voltage.
When C2 is not being held charged by current from Q1's emitter, it discharges through VR1 and R3, and when its voltage drops low enough that Q2's gate goes below its threshold voltage, Q2 turns off and the relay drops out.
D3 protects Q2 from the "back EMF" generated by the relay coil when Q2 turns off and the relay's magnetic field collapses.
R4 protects the circuit against automotive power spikes and surges, and should be connected directly across the power (automotive +ve) and ground (chassis).
The circuit's operation is only well-defined with those three conditions on LOW/DRL: low, high, or alternating at 56 Hz with about 60% duty cycle. Other conditions could cause the circuit to behave in a poorly defined way. For example, a slower alternating signal on LOW/DRL could cause borderline behaviour and put Q2 into its linear region, where it would overheat and possibly be destroyed.
All the parts listed here are available from Digikey or Mouser. The relay type is not specified since you already have one in mind. Q2 is rated to switch 200 mA so the relay coil current must not be more than this.
I haven't drawn in the components for the HIGH beam circuit because I'm not sure how you want it to work. If you want the relay to switch on when the headlight is in high beam, the circuit needs a resistor (1k) and diode (e.g. 1N4001) in series, feeding C2 directly. Let me know if you want me to add this to the schematic.