get the overground around Hackney ! It’s pretty!
most of the museums/galleries have free collections (both Tate modern & Tate Britain, national gallery, V&A as already mentioned — V&A definitely worth a visit) . i also love the museum of the home, the photographer’s gallery, & the ICA (ICA has a cinema and brilliant bookshop also, & it’s v near Trafalgar Square so you can combine classic tourist activity with Something more obscure lol!).
I am a big shopping gworl so i actually love dipping around carnaby street and libertys (which is so beautiful inside!).
depends where youre staying etc and if it stays sunny but honestly the quintessential summer thing to do would be : Sit in a pub with a garden !!!!!
& again - as it’s summer - highly recommend finding a gross little club or gig to go to. if I’m bored I just check resident advisor or dice and see whats coming up.
if you’re comfortable sharing which area you’re staying in then I can send more specific recs hehe