LHR is pretty well connected to London.
Cheapest and simplest way for him would be to hop un the underground at the terminal and change from Piccadilly (blue) to Northern (black) line at Leicester Square.
This would be a few pounds vs. several 10's by Heathrow Express + taxi.
If he is going to stay in London, he'd better buy a Oyster card at the first undeground station which will make his travelling by tube/bus easier and cheaper throughout the stay.