I'd love to see anything really that has not been accessible or opened up to the public for a while - anything I have not seen before TBH.
My wife worked waitressing at St Mary's kitchens in the 80's whilst she was studying - it's an eternal joke about her talents whenever we talk about it as she often dropped the roast chickens only for the '3-second' rule to kick-in and for them to be put back on the dinner plates without anyone noticing!
My list though would include -
1) Whitefriars - have been before but a more detailed / private visit
2) Cov Archives
3) The Earl Street pub cellars
4) Drapers Hall cellar / air-raid shelter
5) Any hidden bits of the Cathedral / undercrofts etc.