I learned from a hiking guide that wearing cotton while exercising in the cold is possibly the worst thing you can do. Wet, cold cotton will weigh you down, make you colder, and it takes FOREVER to dry. Stick with artificial fabrics if you can!
Yeah, I made the same mistake with my socks. I put on synthetic running socks, but they left my ankles uncovered. For some reason that seemed unacceptable in a warm, dry, home, so I ended up plunging my foot into the ice-cold river with thick, calf-high cotton socks.
I have a pair of lightweight wool hiking socks I use in winter runs. My feet stay comfortable below 20 F (about the limit of how low I'll go). They're about twice as thick as my thickest running socks, but they work fine for me. They tuck nicely under my tights or pants.