Recommendations for travel socks

Anything wool! Smartwool is a great brand and if you shop at Costco they have a line of wool socks as well. I have not tried to dry those overnight but wool generally dries quickly.

If you like thin (sometimes called "dress" socks), Gold Toe makes socks that are mostly wool. They also make socks that are mostly acrylic, so pay attention.
If you like thicker socks, similar to those called "crew" socks, then Smartwool or Darn Tough Socks may do -- they're both mostly wool.
If you like thicker "hiking" socks, you can get those, in mostly wool, at any reputable outdoors store, such as REI.
Bottom line: buy socks that are primarily wool. They're both warmer in winter and cooler in summer, and do an incredible job at wicking moisture away from your feet.
And they dry quicker than cotton socks.

Wool is great but it itches. I took Smartwools to the UK a few years ago and after about a week I went into a store and bought several pairs of thin cotton socks. I think I washed them, with other clothes, in a hotel room but it was a multi-day stay so no worry about overnight drying. Smartwool is great for a hike or very cold wet weather, but day after day it got too uncomfortable. Your feet may differ from mine, of course. ;-)

Must disagree with Dick about Smartwool socks.

For me THE BEST socks ever. In fact it's around 6pm in LA and I have on a pair. I know the adage is "nobody walks in LA," but I do. Even today in the rain in order to meet a colleague and catch up.

My love for Smartwool socks are three fold.
1. Comfortable, never itch.
2. Variety of types, hiking, walking, dress.
3. Can wear them for days and they don't smell.

Yes, pricey but you can find bargains.

Try a pair.

I can't wear wool (yes, even SmartWool). Itching, hives, unpleasantness ensues.

I love WrightSock Women's Coolmesh II socks. The ones shown in the link are ankle socks (they hit about 2 inches above my shoe) and I'm in the market for taller socks but haven't settled on any yet. [CORRECTION - they are Quarter socks, not ankle socks]

They are two-layered so that you don't get chafing and blisters even if your shoes slip a little.

ETA: I found out about WrightSocks here on this forum!

MrsEB: What sock would you recommend with the Skechers slip-ons you cite? I'm adding those shoes to my arsenal for Italy in May.

MrsEB: I was out and about this morning and decided to try a pair of Smartwools. I'm about to take a walk with the Skechers and see how the socks feel.

Darn Tough Socks hands down. About same price as Smartwool. Made in Vermont. They have a lifetime guarantee if you get a hole. They wear better than Smarwool. They have a wide variety of styles: low, hi, no show, thick, thin. Plain, striped, patterned. Totally awesome sox. I Wear then all year. These have been family tested. I buy wool sox for 13 family members for Christmas. Originally bought Smartwool, the first year I got Darn Tough, I knew I would not go back to Smartwool. The family agreed!

For cool or cold weather, the absolute best is Falke's part merino knee sock. I've used them in winter in Kentucky, early November in Munich, early October in Russia, and early April in Amsterdam. They arw midweight but warm, come with Left and Right feet and stay up perfectly without the band under the knee feeling tight. You buy them on Amazon for about $22 per pair.

I also recommend Thorlos brand socks. They have different kinds for feet that hurt and those that don't. Nice padded toes and heels. I like the crew socks for women for cooler weather and the shorter ones for warmer weather. They're a US company and had great customer service when I wanted to return some. Take note - if you get the thicker ones, it can affect how your shoes fit. Get the socks first, then go shoe shopping. They probably won't dry overnight, though. Mine dried in about 24 hours.

Merino wool socks (like Smartwool and Darn Tough) should not itch unless you are allergic or super-sensitive to animal fibers.

If you want an alternative in a thin sock,,we like Pearl Izumi cycling socks. They come in men's and women's sizes, and black or white. I turn them inside out so the logo doesn't show.

https://www.amazon.com/dp/B002WRJ24Y/ref=twister_B003BNZPIK