If you’re looking for eccentric and unusual, then Sir John Soane’s Museum or the Hunterian Museum might appeal. The Horniman Museum is not so central, but a lovely place to visit on a sunny day.
If your other half wants to see the Tate Modern, then maybe you might want to try the Saatchi Gallery too. If you’re coming in the next few weeks then the Royal Academy Summer exhibition is always good, and the Serpentine Pavilion normally worth a visit.
There’s a Carsten Holler exhibition on at the Hayward Gallery until early September that looks like it might appeal to kids, got some huge slides and a hang gliding type contraction on one of the terraces. Then, as above, a walk along the Southbank is always nice. There’s normally a food market near Festival Hall, and you can walk across to the Tate Modern.
And if you really want hipster hangouts: Broadway Market/London Fields, Stoke Newington Church Street/Clissold Park, Dalston Yard/Kingsland Road, Boxpark/Shoreditch.