Lived in London for three years. Great questions. Restaurants are hit and miss. If you like pub food, that might be the way to go for a couple of meals since you will only be there a few days. The Indian food is great in London. If you stay where you mention in your post, all sorts of great walking around there. The London Eye is NOT overrated. Tower of London takes a really long time. Some people like it, but I always felt like it wasn't worth the wait. The British Museum is excellent and walking around Covent Garden is great during the day. You should definitely see Buckingham Palace. If you want a more village, old school feel, head up on the Tube on the Northern Line to the Hampstead stop. It's beautiful there...winding side streets and the like.
As mentioned by LeMoyneCuse, have small umbrellas with you at all times, as it could rain on the sunniest of days. It is usually more of a drizzle or mist though.
Lived in the UK for a bit as well.
Concur with a lot of this. The Tower of London is pretty decent, but sucks up a lot of time. I'd rather do a day trip to somewhere like Cambridge than spend hours at the ToL.
London County Hall is a great location - underground is accessible, and you can walk to a lot of great sights with relative ease. I stayed at St. Ermins (Marriott) last time I was there, liked the location - easy walk to where you will be staying. Not sure if one is cheaper. Some of the rooms at County Hall are tiny, no?
All the museums are great - Churchill War Rooms are amazing. The Natural History Museum (love, love love this building) and the V&A are great, but put you a bit out of the way if you're looking to stay central, still you're close to Harrods at that point if that's your thing, and Kensington which is decent enough - The National Gallery is great as well and Trafalgar Square is right near some great theatres if you want to catch a show.
Check out some of the markets - Borough, Camden, all kind of fun and quirky for food and random crap - and cheap eats that are kind of decent if you're not a foodie (mostly food in those two). I don't know, I used to love the markets, so I have lots of thoughts on those. If you're a Harry Potter nut, Leadenhall is the Harry Potter one. Some of them are sort of dull now - Spitalfields, for example used to have some fun shopping, but seems to have gone to crap. Etc...
What can you skip? Nothing. Just try to pack in as much as possible. I wouldn't do the St. Paul's tour if you're thinking about it.
Big Ben is going to be a disappointment. It's still covered up, no?
Download a good tube app - the route planners on the one I have always works great.
