This one is easy:
Ex Officio Give-N-Go boxers.
They’re comfortable, lightweight, and dry very quickly.
They’re also durable. Up until just a few days ago, I was still on my very first pair of Ex Officio boxers bought in 2005; one of the very first articles of quick dry clothing I bought when starting my experiment with super light travel. The elastic waistband finally gave out after having worn them for 12 exciting years of globetrotting around the world. You bet I replaced them immediately, and I can confirm that the quality has not declined a bit over the last decade.
Patagonia also makes some excellent travel underwear made of Capilene, which is their proprietary version of high quality polyester. I own several pair of these (bought a bunch when they went on sale), and they’re great for travel. Between the two brands, if I had to pick one I would still go with Ex Officio simply because I think the material feels better, but you can’t go wrong with either. Both get my highest recommendation.
Ex Officio also makes several versions of underwear for the lady adventurers. They’re made of the same material, though I can’t personally attest to any of those given that I’ve not tried them on.
*Addendum 8-1-2018: not long ago, Patagonia changed the design of their Capilene boxers. I’ve not worn the new versions, but there are many reviewers who are unhappy with the change. Definitely try them on first before you decide to buy.