London in 5 hours

Five hours is cutting it close, but if you really want to get out of Heathrow, Windsor (home of the castle) is a better choice. It borders the airport and is a short taxi trip away.

Big Ben, Parliament, and the London Eye are all close to each other

Close to each other, yes, but not close to the airport. After going through passport control and getting out of the airport (which could be a long time or short, you just never know), the trip into London city center takes upwards of an hour. An hour back, check-in and security again (again, no idea how long that might take), and getting to your gate by boarding time cuts your visit in London to maybe 90 minutes. For me, that would be 90 minutes of stressing over whether we'd get back in time. My vote would be for Windsor, too. At least you'd have time to walk around and enjoy it, not just drive by.

Go to Windsor, it's a fifteen minute taxi ride from the airport. It's a great place to visit.

Windsor Castle is about all you'll be able to do. The Underground into Central London from Heathrow takes about 1 hour. You will not have enough time to do much elso (especially not Big Ben, Parliament etc..)

I hate to sound like a broken record...but...Windsor is your best bet. Not enough time for London itself.

Oh, yes, Windsor is wonderful. You will love it.

Repeating the above, you'll barely reach central London before you immediately have to turn around. We're in London right now and public transportation is not as fast as you might think. Yes, can come into Paddington station on the Heathrow Express in 20 minutes but Paddington is nowhere near the central part. The Picadilly tube will take an hour to get to the central area. I would suggest not attempting a quick visit to London.

Ditto-- Windsor is the way to go-- I gave same advice to Son it worked well for him

Windsor and Eton have a hop-on hop-off bus that covers a few miles in that area. The top is open. The entire loop takes about 45-60 minutes, with descriptions in English broadcast on the bus.

You may want to rethink leaving the airport. A friend of mine just connected in London and said it was a nightmare. They barely made their connection and had about a 2.5 hrs. One relative missed her connection, and another relative who connects in London alot made sure he had at least 5 hours.