I crocheted several hats for my son while pregnant but he didn't fit any of them because he was born early at 34 weeks just over 5 lbs. When I took him home he was only 4 lbs 14 oz.
I followed this tutorial for the Crochet Cluster Beanie.
The Crochet Cap is just double crochets with a brim. It's similar to this tutorial (adjusting yarn and hook size as necessary).
The hats are so teeny that I was able to finish one while nursing (20-30 mins). I'm sure it would be faster if I had both hands free. I had to work around nursing but it wasn't too bad...
10 day old Dustin wearing the Crochet Cap
10 day old Dustin wearing the Crochet Cluster Beanie