I went to YHA London St Pauls and it was GREAT! http://www.hostelworld.com/hosteldetails.php/YHA-London-St-Pauls/London/6977

I'd highly recommend it. It's safe, clean, and very community-like in the common area. It's right next to St Paul's Cathedral which is in the middle of the touristy district which is pretty dead at night, but it's walkable to fun things at any hour. 

Close to there is a pub called "Ye Olde London" That I really liked.

Yes - Check out the hostel on Great Portland Street near Trafalgar Square, Mephisto Picadilly Circus & Leicester Square.  It is about as central a location as you can get in London - sort of like Times Square area in NYC.  Lots of tube & bus options and you are right off the Thames River.  It is a great neighborhood.

Hi, 

I am not quite sure about hostel but we found on our first visit here that the airbnb site was fantastic. It basically allows you to access people's places whilst they are away, if you want the full place to your self or grab their spare room. Worth having a look. 

Grand Plaza Bayswater Apartments is near Hyde Park and the underground. We had a large room with a kitchenate. The were groceries and eating places nearby. The place has an efficient and reasonable over night laundering service which we needed over 2 weeks in France. The place is asprawling and a bit of a maze but we liked it. The price was good. They have a website or you can use booking.com. Get a schedule of London Walks which has a website. We love guided walks in all the cities of Europe. Try the river tour at sundown.

Try Airbnb.com for local accommodation! 

I'd have a look at the YHA hostels first, the only one on this list that's not in central London is London Lee Valley.

http://www.yha.org.uk/places-to-stay/london

Tried airbnb.com ? Run a London search for your group size, some options will be cheaper than hostels and offer great advantages.