It improves the noise margin of Q1. There are some relay modules without this resistor and many complaints about random false triggering like [here][1]here.
It's value is not critical, low enough to have a better noise margin but high enough to leave Q1 a good base current.
The base current can be approximated as the difference between current through R3 and current through R41 considering Q1 Vbe ~= 0.8V
Ir3 ~= Vcc/R3 = 24V/115kohm = 0.2 mA
To estimate the minimum base current required we use the coil/collector current which is less than 17mA for a G5LE relay and Q1 amplification factor that is 50 worst case for MMBT2222, so the minimum base current should be 17mA/50 = 0.34mA
Now we cam see that whoever designed this took a Q1 amplification of 17mA/0.2mA = 85 which is not safe.
Just for exemplification let's take a 150 minimum Q1 current amplification. That leads to a 17mA/150 = 0.11mA minimum Q1 base current. The difference of 0.089mA up to 0.2 mA can go through R41.
With a 0.8V Vbe the result is R41 >= 0.8V/0.089mA = 8.9kohm
Update to show an example with a maximum value of R41.
If there is a leakage current from opto output , Iopto, this will translate in a permanent current through Q1 which can damage Q1 or maintain the ON state of the relay that requires much less current than the nominal current.
Worst case Q1 I collector is Iopto * maximum amplification factor of Q1
For 100nA leaqkage current and max beta = 300 Ic will be 30uA
For this relay the "must release" current is specified at 10% , 1.7mA , much higher than the 30uA above.
The power disipated on Q1 due the leakage current is 30uA*24V = 0.72mW, also in Q1 parameters.
To make sure Q1 is closed we need a voltage less than 0.5V on the base. With a 100nA current that leads to 0.5V/100nA = 5Gohm maximum value.
Obviously here we won't go that high for R41, it was just a theoretical example. [1]: Why relay module occasionally switches by itself ON?
