London for two and 1/2 days...

We loved Victoria & Albert Museum and the British Museum. Also Harrods & Selfridges on Oxford Street for a little retail therapy. Two and a half days is not nearly enough but enjoy!

I'd suggest looking for a place near the south bank of the Thames, between Westminster Bridge and Tower Bridge. That's most central to your three destinations. But you'll inevitably be using the Tube or buses to get around, and any place near or within the loop made by the Circle Line will give you access to the whole city. This site has lots of info about the transit system.

Here's the website for visiting Buckingham Palace, which you can do in August and September when the queen's up in Scotland. Try to get timed tickets online at least a few weeks ahead. I spent several hours touring inside and then took a shorter garden tour.

I think you can book a time for the Eye too, but I don't know how necessary that will be in September. Booking too far ahead puts you at risk of bad weather for this site where the weather will make a big difference.

Try to get to the Tower when it opens in the morning. If you're interested in the crown jewels, go straight to them before the crowds get too big. If not, or later, take the Yeoman Warder tour and spend some more time exploring. You could spend half a day easily.

What else should you take in? Depends on your interests and energy level. Based on my interests, I'd pick Westminster Abbey, the Churchill War Rooms, and the National Gallery as priorities -- all within a short walk from each other but all deserving several hours. Your interests may vary. Check the "Explore Europe" link in this website for Rick Steves' London priorities -- which I think most on this board would consider well chosen.

I'd suggest avoiding Air B&B for such a short stay, you don't want to spend any time trying to meet up for key pickup and return. Buckingham Palace is only open for a short time for a visit, you'd want to check your dates to see if it is open then. The Eye is weather dependent. I'd allow at least 3 hours for the Tower of London, it is expensive so get your money's worth.

I second the Southbank... this is where we stayed in a condo this pasts summer and it was perfect.

We walked everywhere and there were some tube stations near by

You have lots of time. Get a guide book to London and make a list of things you'd like to see. Circle them on the map(s) provided in the guide book, which one hopes will also show underground (tube) stations. I've spent quite a bit of time on the South Bank during my current trip because of the theatres, the craft shops and the Imperial War Museum.

Most visitors with only 2-1/2 days would be focusing on other priorities, and the South Bank is not particularly well connected to the rest of the city by underground/tube. I have my doubts about the area's practicality for this trip, but it all depends on what sights you want to see in London.

It's very helpful to stay somewhere that is very close to an underground station served by at least two lines. That will reduce the need to make multiple transfers on the way to your sightseeing targets. The underground trains run very frequently, but some of the transfers require quite a lot of walking and multiple flights of stairs, leading to rather lengthy trips that would be a pain in the neck with so little time in the city.