One Day: Science Museum 100%. If he likes natural history, that too. And while you are in the area, go into Hyde Park/Kensington Gardens, hand feed the squirrels and parakeets (if he likes animals) have tea in Kensington Palace tea rooms, maybe take a boat out on the Serpentine.
Another Day: If there's any specific things he wants to see in British Museum, go to those. The whole place is daunting, but a quick hour to look at mummies, treasure hoards and the Rosetta Stone followed by an ice cream is ideal. Then head down to Covent Garden/West End to watch the buskers, do some shopping and go to a matinee show followed by dinner in Chinatown.
Another Day: I'd definitely take the Clipper river boat from London Eye to Greenwich, maybe visit Cutty Sark or Maritime museum if he likes boats, Observatory if he's interested in weather, then round trip on the cable car, foot tunnel under the river, and ride the front of the driverless DLR railway. One route is a bit like a fairground ride.
If Greenwich doesn't interest him, you can jump off the boat at Tower of London and explore it then get the boat back to the West End.
If he's interested in politics, you can ask your own MP for a free tour of parliament. Tell them what days you are in London and they will arrange for someone to show you around. Otherwise you can join a tourist tour.