I stayed at the Regent Palace in Picadilly Circus when I went. It's old, the rooms are small, the carpet is threadbare, the furniture is tacky, the TV's are tiny, but it's DIRT CHEAP.
I think I paid like £45 a night. That was almost ten years ago, so I'm sure prices have gone up, but it's still gotta be a smokin deal.
The neighborhood is safe, and you're literally across the street from the Tube Station so it can't be beat. Plus, the lobby bar there rocks because it's full of cheapskate travelers from around the world getting their buzz on.
If you want a classy, new ritzy hotel this isn't for you. But if you want a place to crash for cheap while you explore London, you'll love it.
http://www.regentpalacehotel.co.uk/