Some guys use chainsaw oil or 10w30 and thin it out with mineral spirits which help the thicker oil to flow and then evapourate leaving the remaining thick oil in the chain. I've been using pedro syn-lube for wet, it's ok.

You definatly want something a little more sticky because road slush and snow will wash it off quickly and if it isn't "clingy" then it won't stay on the chain for very long. If you don't keep the chain slathered, salt water will get in and rust it out. I swear you can see the chain rusting somedays