Last weekend, I was in a Birkenstocks store, where I saw Eco-Friendly Socks. They are 48% recycled plastic bottles, combined with cotton and spandex. They were thick and chunky, like all Birkenstock socks, and they felt nice and soft.
If those are too chunky for you, I've also seen these organic cotton socks and EcoWash Merino Wool socks.
The only unfortunate thing about these is that they cost more than you might pay otherwise for a pair of socks. I think in the past I've typically paid about $1.50 for a pair of socks. However, besides being non-organic cotton, my cheap socks tended to get holes in them very quickly. All of these ones here look like they are higher quality and might hold up better.