Please see attached switching circuit.
Why does the n-channel mosfet switched with the test LED test circuit but did not trip the relay? I don't have a flyback diode, but that should not cause a switching problem.
Please advise. Thank you.

 

The position of your pull-down resistor (R2) is creating a voltage divider with the gate resistor (R1). R2 should connect to your circuit on the left side of R1, not the right side as it is in your drawing. As shown, the voltage getting to the gate of the MOSFET may be too low to turn it on completely.

 

The data sheet tells that 5V Gate Source is not enough voltage to get a good turn on.
6V is also low but will work. I would like to see 8V.

Put a diode across the relay.
Change R1 to 100. (value not critical but much less)
OR
Change R2 to 10k. (value not critical but much more)
The idea is to increase the Gate on voltage to 7 to 9 volts. I could be when the relay is on the battery voltage drops to lower than you think.