Maybe an obvious choice, but the Barbican is always good. My favourite space in there is the curve gallery, last time I went was for Cory Arcangel's 'Beat the Champ'. I love the architecture, although not many people do. I think it was voted 'London's ugliest building' in 2003.
SOAS university has a small gallery space dedicated to exhibitions about Asia, Africa and the Middle East. There's a small but beautiful Japanese zen garden on the rooftop (currently closed for refurbishment)
Apart from the fact that its building is worth a visit by itself as it's the only purpose-built stable that functioned as a horse hospital within Central London that is still intact, it also offers some of the best underground and avant-garde culture in London. Aside from exhibitions, film screenings and music events it also houses 'The Contemporary Wardrobe Collection' which features vintage street fashion, couture items and accessories to the Film, TV and Fashion industries.