How do I determine how much RAM I actually need?

Due to space constraints, I need to purchase a laptop to replace my desktop computer. However I don't know whether I need a laptop with 8GB or 16GB of RAM. My desktop currently has 16GB.

What is the easiest way to run something in the background to determine my typical RAM usage over a period of, say, a month? My goal is to have enough data to base my RAM purchase on real-world usage of my desktop.

I've seen a few articles describing how to use Performance Monitor to do this, but they all seem to suggest different things to monitor. Several of them also talk about measuring my swap, which I don't believe I should be interested in.

look at the commit charge value. ProcessHacker shows the peak value in the system information window:

This shows the largest amount of memory that you needed. Now look if 8GB are enough for your daily usage or not.

Given the way software and OS RAM requirements continually expand, my method of sizing RAM in a new computer or motherboard is simple: look into your wallet, and buy as much RAM as you can afford. It won't spoil if you aren't using all of it immediately, but you will surely need it in a couple years (and as you noted, it's soldered into the laptop you're looking at, so you can't readily expand).