Not been there for years, but there is a large mirror in Betty's which is covered in scratched on names - apparently, when Betty's was a pub the barmaid would give the men from the RAF her engagement ring to write on the mirror (my FIL's is on the somewhere).
It's some years since I've been though. The York Dungeon was ok but one part of it was very disconcerting/uncomfortable, which was a young child's cries and sobs coming through on a speaker.
My first comment was a muttered "Which clown brought a young child in here?".
If it's a nice day just walk by the river or walk along the castle walls. The Minster is impressive if you like that sort of thing and gives a good view of the city. And we always enjoy Betty's tea rooms- if you go upstairs it's quieter and the old fashioned interior reminds me of the inside of an old cruise liner such as Titanic. There are lots of lovely tea shops and restaurants.