Museums. National Gallery and National Portrait Gallery at the very least. Free entry.
Walk around Kensington Gardens, see the palace, the Peter Pan statue, the Diana fountain and then maybe head across the park to the Royal Albert Hall for a backstage tour.
St.Paul's Cathedral - it's worth the £18 to walk around in awe and climb the winding stairs all the way up the iconic dome. Westminster Abbey would be another £18 but alternatively you can just attend a scheduled service and have respectful glance around at no cost.
If you're a Harry Potter fan, you should see the Warner Bros. Harry Potter Tour, at the studios where the movies were actually filmed, real sets and all. If you're a real fanatic, also go to King's Cross Station to see Platform 9¾.
If you're a history buff or interested in the Royal Family, take the train (an hour's journey) to Windsor Castle and take the tour, and you'll also see the graves of Henry VIII, and numerous other monarchs at St.George's Chapel. If you're there in June, you can apply for tickets to the Garter Day Parade where the Queen and practically the whole Royal Family strut around in their finery along with the "great and the good".
If you plan on seeing Buckingham Palace, it's best to walk down the mall from Trafalgar Square - by taking the tube to the nearest station you approach from behind which gives a less spectacular first impression.
Beyoncé is not an ally. Actions speak louder than words, Mrs. Carter. #Dubai #$$$