I LOVE going in January! We've been several times during the second week--sadly, I can no longer take my kids out of school.
Okay, PROS: No crowds, short lines. While you'll still need ADRs for, say, Cindy, we walked into most restaurants. If you have "Do not miss" favorites, make the ADRs, but you should be fine.
CONS: Short park hours, some things closed for refurbishment.
Weather: Can be as low as ~35 degrees, esp. morning, esp. EMH mornings. I had my kids wear fleece sweatshirts on the trip down, rather than coats, and we did wear them occasionally during some trips. I would also recommend you bring a pair of long pants per person, and perhaps a second sweatshirt. Another thought might be thin gloves or mittens--nothing fancy, and certainly not for all day, but the mornings might be nippy. We did swim a few times in the afternoons (high ~65). Funny, though, you'll see castmembers in winter coats with fur collars when it's 45 degrees--this is a hoot for those of us from up north.