I did just this last week - I work near Heathrow and live in London and I missed my last train home. Being a girl on her own and too poor for a hotel (I was just about ready to get a US trip, which I've now bought) I opted for the safety of the airport. I arrived at 1am and plumped for Terminal 2 only because it was the closest to the Bus Station.
All I can say is that I wouldn't recommend this one - please discount it as an option. Not much room, and you'll be kept up all night by the sounds of the floor cleaning machines. People start arriving here at 4am for their morning flights and it gets pretty noisy then.

I'm sure someone will be able to give soundadvice about where is a good place to stay the night. But from my own experience - avoid Terminal 2. It's rubbish! (I was just too tired - and vulnerable - to wander around and find somewhere better, besides most of the good seats - i.e. the ones without arm rests you can lie on - were taken by then, so there didn't seem much point).

Best of luck!