Eger, Veszprem or Visegrad?

They will probably be fascinated by the mummies in Vac, which were found under the main square a few years ago during a renovation. Vac doesn't have a castle, but it's a lovely town and only a 25 minute train ride from Budapest. That is, if you take the correct train. There are three: one that goes along the river making express stops and takes 25 minutes, one that goes along the river making local stops and takes 45 minutes, and one that goes inland making local stops and takes 90 minutes. Guess which one I wanted to take, and guess which one I ended up on by mistake.

I saw Eger in the rain and wasn't taken with it, but again, the kids should like the camera obscura, which lets you see over the town.

I haven't been to Veszprem or Visegrad.

Vac is very interesting, but it is not really a place for an "active family" unless you drive and stop and see the bears. Eger is a pretty tough day trip, Veszprem also doesn't seem to fit your bill. Visegrad is a bus trip vs the train, but fills your bill better, especially if you hit it on a day when there is a performance (guys in period costumes). Miskolc has a very classic looking and interesting castle http://www.diosgyorivar.hu/en but it's about the same time to reach as Eger on the train. But if you made it an overnight trip, stayed at the Palota Hotel https://hotelpalota.hu/, you could see the caves http://bnpi.hu/oldal/anna-barlang-136.html and http://bnpi.hu/oldal/szent-istvan-barlang-135.html and ride the old forest railroad http://www.laev.hu/. I would do a driver up, and the train back. The closest storybook castle that comes to mind is Bojnice in Slovakia. About 3 hours in a car. Slovakia has some stunning castles.

Sorry, not much more help, but the Hapsburgs blew up most of the Hungarian castles. Lots of great castles in the Czech Republic and Slovakia if you are going that way.

OH, and I forgot Vajdahunyad Castle in the city park. A recreation but interesting and it seems there is always something going on.

http://www.vajdahunyadcastle.com/

Ran across this that might be interesting for you: http://www.planetware.com/hungary/budapest-surroundings-h-ps-budsur.htm

Thanks, everyone, for your ideas! I really appreciate it.

We're home from our trip so I thought I'd give an update. We decided to go to Visegrad because it was closer than Eger. We ended up taking a train, which takes about 40 minutes from Budapest. From there we took a ferry across the Danube (it's timed to leave about 15 minutes after the train arrives). (We also could have taken a boat up the Danube from Budapest but it doesn't run on Mondays, and we were going on Monday.) We could have hiked up to the castle but we ended up taking a taxi for about $1 per person (there were 8 of us). It was a beautiful, clear day in June. The views were amazing. There were extra paid activities at the castle like shooting bows/arrows and a short horse ride. Near the castle a short walk uphill is a bobsled hill with summer and winter bobsled runs. We also saw signs for a zip line course, but we didn't have time to try it. We were so glad we picked Visegrad to visit. This site gave us some good info for planning. http://featherandthewind.com/2016/11/21/budapest-to-visegrad-hungary/

Thanks for the update. It's always very helpful to have feedback.