My birthday is at the end of January, so I have been there for a birthday trip a few different years. I would say it's one of my favorite times to go. Crowds are low and the weather is usually just fine. It may rain some but as long as you are prepared for it, it isn't a big deal. More rides will be down for refurb compared to the summer, that's really the biggest downside. If you do need to change your dates, I would suggest pushing the trip into February rather than going earlier in January. In particular I found that DLR was practically empty one Superbowl Sunday (keep in mind that in CA the Superbowl airs in the afternoon, not the evening). That evening after the game crowds really picked up, but during the game it was sprinkling outside and we felt like we had the place all to ourselves...fond memories!
Anyway, late January is a good time to go. After dealing with the crowds and humidity this past summer, I think I am going to stick to January or February for my future DLR trips.