London - August, 3 days - best place to stay to see main sights

Because "the main sights" are quite spread out, you're going to be traveling around a lot, probably by the Underground (the "tube"). So you can make life easier by looking for a hotel near a tube station, preferably in zone 1 or maybe zone 2 (you can see the zones here: https://tfl.gov.uk/maps/track/tube). On a recent visit to London we stayed in a hotel in the Earl's Court area. There are whole streets lined with hotels right near the station, and most of them should be comfortably within your price range. (We stayed at one called the Lord Kensington, but I have no reason to think that it's any better or worse than any of the others.) I thought the location was excellent, with easy access to a couple of different tube lines. The hotel rooms themselves are very small, but it's hard to avoid that in London.

khbuzzard brings up a good point with regard to staying close to two different Tube lines. I've stayed twice at Vancouver Studios in Bayswater. I like the hotel (the rooms have kitchenettes, with a tiny fridge and cooktop, plus utensils), and I like that it's close to the District and Circle lines, and also the Central line. Different Tube stations and sometimes whole lines are often closed for "planned engineering" (maintenance), so you want to have options. I recommend checking the Transport for London (https://tfl.gov.uk/) website every morning to see if there are any station or line closures.

Thanks for the suggestions and great tips - I don't think I'll ever plan European travel again without checking the forums for the first-hand experience from everyone on here. I'm finalizing quite a few plans now, so much appreciation to you all for the help!

Travelodge or Premier Inn fits with your budget and they are everywhere in London. Nothing fancy, but always clean and reliable.

Lancaster Gate is a fine suggestion, not on top of any tourist sights but 10-20min tube ride away from more or less anything.