Tuesday, May 24th we land at Heathrow (from Barcelona) at 10pm. Our flight home to Atlanta leaves on Wed the 25th, at 3pm

So, the 4 of us need a place to stay overnight. Wife and I (mid 50s) would likely just stay somewhere in or near the airport. Any hotel suggestions? OK to be quite budget, as long as clean.
But, our son and friend (young and eager) want to consider trekking into London, even if they stayed awake all night and just saw the sites.

Is there a convenient train or tube into central London? Can you catch it at Heathrow (we will be arriving at terminal 3, but leaving from terminal 5). What is the cost, and travel time? Where would it "put" them, once in London? Any connections?

I know it likely sounds nutty, to try to tackle that, but they are young and energetic, and she has not been anywhere in Europe before. They feel that, even if they could stroll along the Thames and view Tower Bridge, Big Ben, the Parliament, it would be worth it.

Any suggestions, please post them, and thank you for your help and your time.
EK

Given your late arrival, I suggest you pick a hotel very close to a Piccadilly line Tube stop (ideally between Piccadilly Circus and Holborn, inclusive) but take a pre-booked car service to the hotel - for 4 people, a car service can be cost-effective. The next day, you can either catch the Tube all the way back to Heathrow (slow, but your flight isn't until 3 pm, so you have plenty of time) or take the car service back. Remember, the Tube doesn't run all night, so being centrally located will be a big help, although your son can take the (shudder) night bus.
http://www.heathrow-airport-guide.co.uk/transport.html

Anyway, that's my 2p based on years of travelling frequently from the 'burbs near Heathrow to London

From London, you can either take the Tube or the train. There is a station for both at T3 and T5. The Tube will be the Picadilly line, so any stop along this would work well enough, or they could transfer and the world would be their oyster.

The train leaves you at Paddington. Personally, since I don't usually have much occasion to be near Paddington, I don't find it much more convenient than the Tube, but I have done the train to a taxi before and it works well enough.

They would have some time in the morning, but would probably need to leave London by noon for a 3pm flight.

The Tube takes about an hour to get to central London from Heathrow - though the last train all the way into town appears to run from T5 at 11:42pm. So if it will take awhile to get bags and all that, they may want a car service instead. (There are separate stops at T3 and T5.)

I wouldn't bother with the train (Heathrow Express or Connect) - getting or staying more central than Paddington is essential in this case, if you're not staying at LHR. Otherwise it just misses the point. And the cost of the train+taxi would add up quickly for four people. You might as well get a car service if you're willing to pay that much.

Most of London is quieter than you might think at night (the Soho area being an exception), but if they want to stay up all night, walking by Big Ben, Parliament, the Eye, the Tower... always an option and beautiful at night. If that's their plan, I would take the Tube to either Piccadilly Circus or Leicester Square and walk from there. If they really want to see the Tower, it's not near a Piccadilly stop - they'll have to transfer to the Circle/District lines (if available that late) or take the night bus. Or wait till morning - the Tube starts back up around 5am, I think.

For a 3pm flight, you'd want to be back on the Tube between 11-12, depending on how much time you want to have before the flight (at least two hours, some people/airlines say three). So if you and your wife wanted to join the "kids" in central London, you won't have too much time to see things like museums. But if you're centrally located and can get up and at 'em early, you can walk around and see a lot before you leave. The most centrally located Piccadilly stops for your purposes would be between Piccadilly Circus and Covent Garden (Green Park and Holborn might be OK too). You can walk to a lot from hotels near those.

How would I arrange a "car service" for that trip for 4 adults, from Heathrow into London, at 11pm or so?

Thanks for all the answers so far!!

<i>How would I arrange a "car service" for that trip for 4 adults, from Heathrow into London, at 11pm or so?</i>

http://www.justairports.com/

You can book online or call. I would say that they are the most recommended car service on this board and seem to be pretty reliable.

Ray Skinner is another who gets recommended, though I don't have his information. I've used Just Airports - they were not very responsive via email, but showed up at the airport as I had requested and were otherwise fine. You'll give them your flight info.

Here's Ray Skinner's contact information. http://www.london-transfers.com/

I've used both services, I think Ray's is a bit more expensive but he's good. We did have a glitch with our return with just-airports once. I would use both again, but reconfirm your return with the main office--even if they've just dropped you off the night before.

This could help out with the hotel choices... www.heathrowairporthotelsaccommodation.co.uk there's filters for star rating, budget and distance from the airport so you should be able to find a hotel that meets your needs.