We recently had the chance to visit the Museum of Natural History with our three-year-old son. He was duly impressed (though only time will tell whether the museum makes the same impression on him as it did on me).
At age 3, it turned out that he is a bit too young to enjoy the Planetarium's Journey to the Stars show (which was amazing, though a bit too loud).
The dinosaurs were more accessible. In the years since my last visit in the early 80s, the museum has re-positioned the dinosaur fossils to alter pose and posture, taking into account the most recent learning concerning the dinosaurs. From a 3-year-old perspective, I suspect the size of the fossils makes the most significant impression.
The museum now has at special exhibit on the human brain. Great exhibit for older children (perhaps 7+) and adults. My son felt the exhibit was a bit too dark, and so asked me to hold him through the entire exhibit (which I love doing anyway).
If you find yourself in Manhattan with your child, make a point to stop at this national treasure.
--TC Daddy