Peckham, Hackney and Dalston: the weekend London actually lives in
South of the river, Peckham has built a real weekend identity around West African restaurants, modern wine bars and Peckham Rye Park, plus Peckham Levels' cluster of food stalls and bars in a converted car park. North and east, Hackney runs Broadway Market on Saturdays and keeps London Fields busy on any dry afternoon, with Ridley Road's Turkish food close by; Dalston pairs Turkish ocakbaşı grills with a genuinely late-night bar scene. None of these are secret anymore, but none of them are built for a tour bus either — they reward simply staying local for two nights rather than checking off a list.
Southwark: the practical base
Southwark has some of the best transport connections in the city and enough of its own character — Borough Market, the South Bank, Bermondsey Street's galleries and restaurants — that basing a two-night weekend here means most of what you want is a walk or one Tube stop away, not a 40-minute cross-city trip. It's the closest thing on this list to a "safe first choice" for a first-time weekend visitor.
A slower alternative: Little Venice
If the whole point of the weekend is to slow down rather than speed up, Little Venice's canal paths and houseboats are a genuine change of pace inside the same city — a short walk or canal-boat ride from the more obvious central sights, with almost none of their crowds.
What to actually skip trying to do
Don't plan to see the Tower of London, the British Museum, Camden Market, Shoreditch and a West End show in the same 48 hours — that's five separate half-day trips stitched into one exhausting itinerary. Pick two neighbourhoods, one big sight if there's one you genuinely care about, and leave real unstructured time in between. London rewards wandering far more than it rewards a checklist.