Other options might include:
Take a cab to Leamington Spa (only about 10 miles away, so should be much cheaper than getting a car all the way to Heathrow), where trains start running much earlier in the morning, and you can get to Heathrow as early as 9:00-ish.
Or take the train to Oxford (starting from either Stratford-upon-Avon or Leamington Spa), walk the 1/4 mile from the train station to the bus station, where there's frequent bus service to Heathrow. This takes approximately as long as the train via central London but may be cheaper.
And by all means, if you want to take the train all the way there, split your ticket to save money. Most of the train routes have you go to London Paddington and then take the Heathrow Express, and they want to charge you 70+ pounds for the privilege. But you can get a ticket to London Paddington for about 30 pounds, which saves you money even if you pay the walk-up fare for the Heathrow Express. (And since you're traveling on a weekend several months from now, you can pre-book your Heathrow Express tickets and save a lot of money there too.)