How would I determine the thevenin equivalent resistance?
You can't directly, at least assuming there is nothing between the arduino output and the microcontroller output.
Microcontroller manufacturers don't spec output resistance usually, but they do specify the worst case voltage drop under some load. For example, for a micro running at 5 V they spec might say the minimum guaranteed logic high output is 4.0 V at 20 mA sourced (I'm making up these numbers as examples only, you need to check the datasheet). Assuming the output when driving high is a saturated FET that can be modeled well enough as a resistor, then this says it drops 1 V at 20 mA. 1V / 20mA = 50 Ω. That's the worst case, so is the highest resistance. The actual resistance can be anywhere from 0 to that value.
However, modeling a digital output as a Thevenin source is only roughly right. You can get some first order useful information from it, but you certainly don't want to use that for any kind of detail. If you find yourself wanting that, then it's a good clue you need to step back and think about the design differently.
It would help if you explained why you want to know the output resistance of the micro.
To start with, there's no single answer to that question, because the pin characteristics are determined by the MOSFET transistors that drive it. These characteristics can vary considerably over process, voltage and temperature.
You can get an approximate value by looking up the DC characteristics of the chip pins in the datasheet, and doing a ΔV/ΔI calculation based on two different pairs of VOH, IOH or VOL, IOL data.
But the better answer is to design your external circuit so that the specific source resistance of the output pin has little effect on its overall performance. If you give us some idea of what you're up to, we can offer more spsecific advice.