[sowtime444]

I personally like Bankside. Very central. Walk left along the river and get to Big Ben. Walk right along the river and get to Tower Bridge. You can't go wrong. The Tate Modern is also right there.

Hilton London Bankside is nice. I'venot stayed there but I've seen the lobby/pool. It's very central. I see $163 on hotels.com for August 1st.

Citizen M Bankside is nearby and also nice. I've seen the rooms and they are very "interesting" - you can press a button on a remote and turn all the lights red for example. I see $177 for August 11 on hotels.com.

To find the cheapest just look for the Travelodge's (which is cheaper than Premier Inn by a little bit) or an older boutique hotel that might be having a sale. There are one each in Bankside.