Central london shopping is a possiblity, Selfridges (a very large department store) is open till 9:30 M-Saturday. I think thursday is one of the later nights shops are open on Oxford Street, but 9:00-9:30 would be the latest.
Like other people have said, not much going on around Heathrow. Do you know what terminal you are in or what airline you are flying? I know you can purchase one day access to American's lounge for like $50 - I would assume you have to be flying with them, but not sure. The one in Heathrow terminal three is nice and has showers and snacks and comfy chairs. A bit nicer than just being out in the open terminal.
Also - I am not sure all airlines will let you check in 12 hours before a flight. There really is nothing interesting in the terminal before you go through security. So if you can't check in, you can't go through security and you will have all your bags with you etc. Might be worth a call to the airline.
You can take the tube into central london, but if it is this Thursday, there is a Tube Strike planned for that day. Often times it doesn't disrupt anything or is cancelled at the last minute. But I would check the TFL website on Thursday and adjust your plans accordingly. The Heathrow express may be a better option. More expensive, but less worry.