You may already know this but on babycenter.com there is a section called "Tools" or "Calculators" where you can predict a child's height. The soonest they will use measurements to plug into their calculator is when the child is 4 years old.
Also, I was told by my pediatrician another way to estimate height. For boys, take the mother's height and add 5 inches and then average that with the father's height. For girls, take the father's height and subtract 5 inches and average that number with the mom's height.
I am sorry I can't comment any more than that, as I don't have any personal experience---my oldest is 8. FWIW, as far as genetics go height or being tall is dominant. So even though your son's parents are average in height, your consideration of the heights of past generations is completely relevant. And lastly, I am the shortest in my tall family at 5'10" and my husband is the tallest in his average sized family at 6'4"---I think those results are very much affected by nutrition because I was a picky eater and my husband eats everything. My husband also heard that if the body undergoes stress (like starving yourself for high school wrestling--that's what he did :0) and then eating again after the season and going through a growth spurt, the growth responds favorably for height to that as well. My meaning---there are so many factors. So for an estimate I would suggest you use the two methods above and then average those and I bet you will come pretty close.
Best wishes.