January is our favorite month to go. We've gone several times during that month over the past years. Our last January trip was just a couple of years ago. We stayed at Port Orleans Riverside and sometimes we were the only people on the bus when it picked us up and took us to a park. The buses weren't crowded to come back to the resort in the evening either.
Yes, the weather might be a bit chilly for the locals but we're from Michigan so just getting away from the snow, ice and frigid cold is great to us!
We usually can wear shorts/t-shirts during the day, and if it gets chilly in the evening we go back to the resort and change to jeans/sweatshirts.
I do remember one evening in the Magic Kingdom when it was cold enough we could see our breath, and I was kind of wishing for a hat and gloves but we still had a great time, and even got to ride Thunder Mountain twice (not getting off after the first ride) since the crowds were so low.
Generally the rides/attractions are either walk-on, or maybe a 5-10 minute wait for the more popular ones.
Plus I think it's the least expensive time of the year to go and stay on site.
I've been to WDW in March and September as well, and didn't care for either of those months. March was spring break for a lot of kids/colleges so it was super busy, and September was also quite busy but still very, very hot and humid.
We'll be going again the last week of January next year, with our DD and her DH and their two kids plus her DH's parents. Can't wait, as it'll be the grandchildren's (ages 6 and 9) first visit!