I had a free hour in my conference day so I walked over to the North Carolina Museum of Natural Sciences. The museum is located a block from the capital and across the courtyard from the North Carolina Museum of History.
The museum accepts donations for admission if you desire and when I hurried over there it was a little after 5pm on a Thursday. The gift shops were closed but the exhibits were still open.
The museum while smaller did have some very nice exhibits. There are the mountains to the seas, coastal North Carolina and an underground exhibit.
The place is not large but would be great for a rainy day to take up an hour and poke around.
Monday–Wednesday, Friday & Saturday: 9 am–5 pm
Thursdays and First Fridays: 9am-9pm
Sunday: Noon–5 pm
More information can be found here: http://naturalsciences.org/