Best for egg sandwiches....the design museum in southwark. Nice dense bread and egg with good mayo and most likely butter. Also does an excellent stem ginger cake. I can't comment on the exhibitions as I last went to one years ago. There is an entrance fee and the exhibitions sometimes look a little dry.
There is a fantastic view of London down towards Tower bridge and beyond, in summer you can choose one of their brightly coloured child style chairs and eat outside. However, one warning, this lunch, although delicious , is not cheap.