My kids had a blast with the trams and metro lines and the boats and the bus that drives into the Danube River. They even enjoyed Beauty and the Beast performed in Hungarian at the Operett Theater (with English subtitles). Just being someplace new and strange was “almost” enough.
Still, there are some things kids like. The list covers tiny kids to over 21; individual and with family
Amphibious Bus: http://riverride.com/
Boat Ride from Szentendre: http://www.mahartpassnave.hu/webset32.cgi?MAHART@@EN@@115@@GOOGLEBOT
Caving under Budapest: Imagine crawling through caves in a country with very few lawyers: http://caving.hu/
The Budapest Zoo. Animals for the younger and great architecture for the older. My wife and I still go every couple of years. http://www.zoobudapest.com/en/must-see/zoo-history
Tropicarium (aquarium). http://www.tropicarium.hu/Tropicarium-Budapest-Hungary
Aquaworld (one of the largest enclosed water parks in Europe) http://www.aqua-world.hu/en/aquaworld/water_slides_and_activities?sid=7379fb4a64670dfc20503ba070198d85
Children’s Railway (meaning operated by children for children and adults). A hold over from the Cold War communist Pioneers youth group. http://www.gyermekvasut.hu/english/chrw_home.php Pretty ride through the Buda Hills, on some occasions by a steam locomotive.
Public Transportation sight seeing routes by Tram, Underground (oldest on the continent) and bus: http://www.bkv.hu/en/our_thematic_routes_with_/
Baths, Spas and Beaches: http://www.spasbudapest.com/
Palace of Arts: Has a number of concerts and activities target a young age groups: https://www.mupa.hu/en/events/calendar/09/2014
Exit and Escape Games: Becoming very popular in Budapest and target at a broad group of ages. This is one link that explains it, then google “Budapest Exit Games” to find more venues. My kids are looking forward to this at Christmas http://www.budapest.com/leisure/special_programmes/daytime_programmes/sweet_escape_-_escape_room_budapest.en.html
Challengeland Adventure Park http://kalandpalya.com/lang/en//
Museum of Transport http://www.mmkm.hu/Honlap/Szorok/INFO_ENG.pdf
Chairlift through the Buda Hills https://hu-hu.facebook.com/libego
Drive a Russian Tank http://www.tank.hu/
Ruin Pubs: http://ruinpubs.com/
Natural History Museum http://www.nhmus.hu/en (voted most family friendly museum in Budapest)
Budapest Underguides I believe has an interactive tour or two for children. http://underguide.com/ and a few tours I would keep my kids as far from as possible.
Center of Scientific Wonders (rough translation), a center filled with interactive science. http://www.csopa.hu/eng/