I'm not a London expert by any means, but did visit this past summer (wrapped around a trip to Croatia). On the first part of our trip, we stayed in an apartment in Marleybone on Baker Street (just off the Marleybone Road). We were walking distance to restaurants, shops, Regents Park, and the Baker Street underground stop.

On the other leg of our stay, we stayed in a hotel ($$$$) in Mayfair. Expensive, but walking distance to Green Park, Hyde Park, Buckingham Palace, etc. etc. I would think that the nearby neighborhoods (Chelsea, Kensington, Belgravia) would also be good places to stay.