@nurbit
If you have time and will follow the link at the start of the topic to dr. Le Hung's page from where I got the original schematic you can find some good information on how this ESR meter works.
Until then I will try to explain in a few words some basic things that will help you understand why some components are used.
To calculate the ESR we use ohm's law for which the formula is V = I * R, for us the ESR is represented by the R from this equation, thus R = V / I, now we have to get these two values from somewhere so we can compute a result.
For this we setup a circuit which will connect the DUT to a known current that in turn will produce a voltage drop on the DUT which we can measure through the analog pin of the arduino, depending on the resistance (esr) we need to measure we might need different current values, the sketch implements just the 50 mA branch, a 5mA branch is available on the schematic and can be easily used switching to the proper pin for higher ESR measurement (in case of lower value capacitors, let's say 10uF and smaller) thus it can be implemented as an auto-range function.
How we obtain the constant current:
For a 50 mA current we have: (I = V / R) = 5 / 100 = 0.05 A. Values are from: for Vcc = 5, this is supplied by the arduino's on board voltage regulator and R = 100 (obtained from R8 + the resistance of the transistor at saturation which should be very close to 0 ohm, thus 100 ohm).
So the reason to use low tolerance resistor in the current circuit is to ensure the value of the current is exactly as expected and calculated.
The 1 k ohm 1% resistor (R4) branch can give us a current of 5 mA (I = V / R) = 5 / 1000 = 0.005 A. I have to mention that the 5 mA current is not implemented in the sketch, so you can leave it out for now, this is necessary if you want to measure higher ESR or higher value resistors.
Some more things about the way we measure: we need to pulse the DUT with a very short pulse so that the capacitor we are measuring (especially if is a lower value) will not have time to charge and mess up the reading, plus charging and discharging fast is needed so that we measure the capacitor with a frequency as high as possible so capacitive reactance will be negligible (this is the reason I tried to speed up the analog read by changing the prescaller).
Conclusion: you can use 5% resistor (you can try to calibrate the device from the variables in the sketch), I guess the A1015 should be good too for a 50mA current, give it a try and see what are the results 