What is the fastest way (cheaper is better, but if unavoidable, costly is agreeable) to reach Cambridge from Heathrow Terminal 3, at night (past 10pm)? I expect quite a few people who reach Heathrow at night will be traveling there, so if someone in London/Cambridge or anyone who's done it before could share, it will be great.
The cheapest (and simplest) way would be to use National Express coaches (i.e. bus) that goes direct from Heathrow central bus station (5 minute walk from terminal 3) to Cambridge. Tickets start from about £15, depending on the day/time and how long in advance you book.
I just checked their website - the first bus in the morning leaves Heathrow at 5 am and the last one at 11:59 pm (although I wouldn't recommend this one, as it's not a direct one and you'll spend almost two hours in Luton airport when changing to another bus). The last direct one leaves at 11:35 pm. Hence at 10 pm you should be fine. The journey takes 2 hours 20 minutes.
Check for yourself: http://www.nationalexpress.com/bp/b2.cfm?id=265500945&ochoice=all
Another option is train, but it's more expensive and there are no direct trains - and it's not faster.
Get the Piccadilly line from Heathrow to King's Cross (~1 hour) then catch a direct train to Cambridge (again ~1 hour). You can check trains from London KX to Cambridge here. The last one today leaves at 00.31 but in general I'd assume it will be around midnight. Provided you get onto the tube at Heathrow by about 10.40 pm you should be fine.
If really pushed for time you could get the Heathrow Express to Paddington then the tube to King's Cross but I would advise against this. The Heathrow Express is ridiculously expensive (£20) and would save you 15-20 minutes at most.