Hi Andrew,
Celebrating your son's 4th birthday at Walt Disney World will be so much fun! You have chosen a wonderful resort at which to stay.
Disney's Grand Floridian Resort & Spa is gorgeous. Also, since you are planning to spend one day at Magic Kingdom, the location is perfect. You can hop on the monorail and be to the Magic Kingdom very quickly!
You are correct, typically mid-January is not too crowded at Walt Disney World. One event that will draw crowds is Walt Disney World Marathon Weekend, which is January 8 - January 12, 2014. While you should be aware that this event is taking place, it should not make crowds too large. You can expect fairly low to moderate crowds that week.
I suggest that whichever day you choose to go to Magic Kingdom, try to arrive as early as you can, about 30 minutes before opening if possible. When you arrive, do the headline attractions that you are most interested in first - rides such as Big Thunder Mountain Railroad or Space Mountain. The parks are typically at their least crowded early in the day, so lines will be shortest in the early morning.
One more idea - do be sure to stop by Guest Relations when you arrive at Magic Kingdom and ask for a free birthday button for your son. It is so much fun to hear cast members and guest alike wish your little one a happy birthday, and sometimes a cast member might be able to sprinkle a little extra magic on a child's birthday.
I hope that this information is helpful for you, Andrew. Have a wonderful time planning your vacation, and please do not hesitate to ask if you have any further questions.