Easiest way to buy a one day Tube pass at Heathrow?

There is a Tube station in Terminal 5. Follow the signs to the station and buy your travelcard there.

The travel card DOESN'T cover the cost of the pricey but oh so convenient Heathrow Express. Two different transport entities, the London Underground (tube) and the Express. If you buy the card at LHR you can use it on the 45 minute tube ride into London. Personally, given you have only a day, I'd jump on the Express to Paddington station (15 minute ride) and buy your travel card at the Underground station located at Paddington. Walk from your train to the entrance of the station. Simply follow everyone. Can't miss it. Red Circle, White background with Blue Line through it is the Underground logo.

I'd buy the tube pass at Heathrow and then take the Piccadilly line into London. I personally find the Heathrow Express saves me very little (if any) time unless my destination is Paddington. There aren't many tourist sites around Paddington, so I suspect that you will want to go somewhere else. That would mean waiting to purchase a separate tube ticket at Paddington, walking to the tube platform, and then taking the tube to your "final" destination. Once you do this, you've eaten up most of those time savings and spent quite a bit on that expensive HEX ticket. With the Piccadilly line, it is a straight shot to many stations close to tourist sites (e.g. Leicester Square, Covent Garden, Holborn).

I'm back from my trip! Heathrow was surprisingly quiet and efficient the day before the Games - I was through immigration and on the Tube 45 minutes after landing! Thanks for the advice - the whole day was smooth, and London just before the games was an exciting place to be.