There's plenty you could try to pack in!
If you're planning on dropping bags off at your hotel (they'll be happy to keep them safe if you don't want to lug them around town), then you'll probably be able to get a local bus from the terminal to your hotel; most of them are in the Free Travel Area. They're mostly shown on
this map [pdf] (but if you can't make head nor tail of it, then if you let us know which terminal you'll be arriving and which hotel you're at, we'll be able to de-mystify it for you.)
Next: getting into town. From any of the Heathrow tube stations (including Hatton Cross, which may be nearer your hotel), you can get a zones 1-6 off-peak one-day travelcard for £8. This will let you make pretty much any public transport trip in London after 0930 - the only significant exception are the Heathrow Express and Heathrow Connect trains.
I'd suggest taking the Piccadilly Line to South Kensington, and having a wander around there. I could easily spend a week exploring each of the
Natural History Museum,
Science Museum and
V&A, so you could have a look at their websites and see if there's anything on in particular that would be worth a look. They're all free too, apart from special exhibitions.
If it's a nice day and you're feeling fit, you could wander up to Hyde Park and rent a bike from the
cycle hire scheme - aka Boris Bikes, named after our loveable mayor. You pick up a bike from one docking station and can return it to any other, and provided none of your trips last longer than 30 minutes, only costs £1 for the day. You can then zip straight through Hyde Park, into Green Park, and stop by to wave at the Queen at Buckingham Palace.
..and if it's not so nice, then you can get the District or Circle line from South Kensington down to Westminster, and see Parliament (hint: Big Ben is the name of the bell inside the clock tower - you can be smug and point that out to other tourists). Wander up Whitehall, and you'll pass Downing Street, plenty of other fairly photogenic government buildings, and shortly get to Trafalgar Square - which isn't square, but is home to the
National Gallery and
National Portrait Gallery (again, both free). Lots of cafes around here too should it be lunchtime by the time you get there.
Other things to consider are the
Tate Galleries (one's full of modern art, the other more 'important' stuff - you can take a boat between the two if you want to see both); as others have said, shopping - or at least browsing; try Selfridges, up at the Marble Arch end of Oxford Street.
And check out
timeout.com/london for some things to do with your evening - if it's theatre, there's a TKTS booth on Leicester Square; if you fancy some comedy there's plenty of small gigs on the go every night. The last tube back to Heathrow doesn't leave central London until just after midnight, so don't hold back!
Oh, and when it comes to dinnertime, make sure you avoid the Aberdeen Angus Steakhouses, which is mediocre and expensive yet inexplicably still attract tourists like flies.