Dear Erin: Of course no itinerary of London can include everything, and 3 1/2 days isn't much time, which makes it harder still....but my suggestions are as follows.
Your arrival day...if you arrive at noon it will be hard to get into central London, get to the hotel, check in much before 2:30. Like many others, I strongly support exercise/fresh air that first day to ward off jet lag. On the other hand, if it is really pouring rain---usually London is just drizzle, but real rain isn't impossible--that makes outdoor activities less appealing. With that caveat, I would suggest walking through Hyde Park and unwinding after the flight. Then, tube/walk to Piccadilly Circus...have afternoon tea at the Wolseley (in Rick's book). Great little tea sandwiches, pastries, scones...you'll have no need for dinner.
Then think about an evening walk with London Walks company rather than the Eye. IMO the Eye is just a ferris wheel, and the view is far less interesting than, say, the view of san Francisco from Telegraph Hill.
Day 2: You won't get much out of the City...think midtown Manhattan. Virtually all of it was bombed in WWII and is modern. Why not start at the Eye, walk along the South Bank to the Tate Modern--do an hour+ there unless you hate modern art...walk across the Millennium Bridge which leads directly to St Paul's. Do that and then perhaps an early dinner followed by theatre. London theatre is even better than New York's.
Day 3 Too much. Westminster Abbey will take until lunch. Why not do Rick's Whitehall walk, do the War Rooms/Churchill Museum on the way, then walk to Covent Garden. Shop and eat dinner. Perhaps another play or walking tour.
Day 4. British Museum. Harrod's IMO is just a crowded department store.
There are lots of other possibilities: Museum of London, National Gallery of Art, and Transport Museum in Covent Garden are all very good.
You'll have a great time..enjoy!
