My favorite hiking spots around Austin were Hamilton Pool, the Barton Springs greenbelt, Pedernales Falls State Park, and Enchanted Rock (a bit of a drive to get out there- but you could enjoy an afternoon in Fredricksberg; also great for camping.) Enchanted Rock is my favorite. I think Inks Lake is in Marble Falls and that is a nice area,
If you wanted to go to some caverns/caves Longhorn Caverns (this is in Marble Falls), Innerspace Caverns or Natural Bridge caverns are nice options.
For BBQ- I would say a trip to The Salt Lick is a must but maybe too long of a drive to Driftwood from Marble Falls.
It's been so long that I've lived there I can't think of museums. There is a new aquarium your niece might enjoy. Is 6 too old for the children's museum? They have a great one.