You can spend the day in London and take the Heathrow Express direct from LHR T4 for £16 return, as long as you return the same day. Problem is you have to buy it after 13h00 I believe. You can then buy an all-day underground pass (at any tube station at the automated machines or manned tellers in some stations) for only zones 1 through 2 or 3 for about £3.

You can also take the tube from T1 after taking the Express from T4 (free), and buy an all-day pass after 09h30 for about £6. 8 hours is enough time to get into the center of London, see a museum, do a little shopping and be back just in time. I've done it before many times, using the tube. Make sure you're back on the Piccadilly at least 2 hours before departure though. Museums in London are all free (shopping at Harrods or Old Bond Street is not). You could make a day of London for as little as £10 including transport and a small lunch (and on Sundays, you could attend services in Westminster or St. Pauls and avoid the usual entrance fee, plus if you're religous...)

I can give more specific suggestions if you want, but I'm sure the natives may have better advice.