Joan Meyer
Google
Even without a ticket, you can climb the hill and take in the beautiful views.
The museum is included with the Budapest Tourist pass (as is the National Gallery and the Libraries) or you can pay for each separately.
St Stephen Hall is apparently a big attraction for tourists, but I much preferred the exhibitions about the history of the area. I found the information about the region's geography and many cultural influenced interesting. I enjoyed their collection of artifacts from prehistory, the Roman era, the Medieval period, Ottoman rule, and up to present day.
I imagine it would also be a good spot for kids - there's a number of educational games that I (a full grown adult) loved playing.