hi all-
i will be getting into Heathrow at 620am next monday and not leaving until 4pm. should i stay in the airport or go into town? if town, any suggestions on how to get there fastest and cheapest (tube, i assume) and what to do? thanks!

Definitely take the tube into the city. Then go on the sightseeing hop-on hop-off bus so you can see all the sights.
10 hours in the airport would be a killer.

Or go to Windsor & see the castle & Eton.

But once you've gone through immigration, check to see what time you need to be back. You might find your free time has magically gone down to nearer 5 hours.

You can get into London in 15 minutes or so on the Heathrow Express.

You may be able to get to PADDINGTON station in 15 minutes using the HEX BUT that doesn't get you to any of Tourist London which requires a further Tube ride.

It's much simpler, almost as quick & a LOT cheaper to get the Tube to somewhere like Hammersmith then change to the District Line & get off at Westminster which puts you in the heart of Tourist London

The fastest and cheapest are not the same. Fastest will be the Paddington Express, 20 pounds I think. Cheapest will be the tube, buy a 1 day 6 zone pass and you can use it unlimited all day and then come back on the same ticket. What to do - wander thru Hyde Park, take the sightseeing bus, go to a fun local market eg Covent Garden,

I would NOT take the HEX. Sure it only takes 15 minutes but as alanRow says - it only gets you to Paddington. From there you would have to queue for a taxi and then have an expensive cab ride to anything of interest, or take the tube from there. It really doesn't save you much, if any, time and costs a lot more. His suggestion of tube to Westminster is good - you step out of the tube station and are looking straight up at Big Ben with the river over you left shoulder. You will be across the street from Westminster Abbey and a short walk from Trafalgar Square or Buckingham Palace

Windsor has also been mentioned. Windsor might be a better choice since you will only really have about 5 hours free. It is a 10-15 minute cab ride or a 30 minute bus ride and you can see the Castle, river, wander around the town, have a nice lunch and catch a cab back

Keep your options open. If the tube is running smoothly on the lines suggested above, it's an acceptable route. But if there are significant delays, then the HEX option is also good. You don't have to take an expensive cab or the tube. You could walk out the station and catch the #15 bus, which runs essentially the same route as much of the hop on tour buses. When I did that with my mother in May, we had tour buses right behind us for most of the ride. Take an upper level seat and you'll have a nice ride through a good chunk of central London.
As for the cab ride to Windsor, I wouldn't bet too much on the ride being only 10-15 minutes. We take that route to go to Heathrow and it's usually a good 20 minutes minimum, even in the morning. Too many weird traffic jams these days (both ways).

Although you don't really have 10 free hours, you do have enough time to take a nice peek at London.

we went to windsor under similar circumstances and had a great time.

there was a bus at the terminal we left our hand luggage at in storage. it was very easy, fast, and a lovely setting in windsor with a boat trip on the thames...then a nice meal outside on a patio at a heavily flowered pub garden.
nice break without the hassle.

i have a trip report from some time back but i have been unable to retrieve it for you.
it gave the bus number, etc.

I am actually in favour of Heathrow Express. It is definitely much faster way to get to central London vs taking the Picadilly line. (Esp if you're arriving Terminal 4, where there is no currently direct tube service.)
Paddington is served by a few tube lines, I see no need to take taxis. You can easily get to Embankment or other places of interest on Bakerloo Line.

On last Sunday it took us 80 minutes just to clear immigration. Customs took 30 seconds. Just something to keep in mind.

< Customs took 30 seconds >

What took so long?

We had a similar layover on our way to Nairobi. We took the train over to Kew Gardens and then walked along the Thames. It was great to get out into the sun and get oriented to the time change.