Is there a cheap way to travel by train from London to Edinburgh?
Stephanie, The cheapest way that I know would be to take the short trip (~4.5 hours) on East Coast Rail from London King's Cross to Edinburgh Waverley. If you pre-purchase your ticket well in advance, considerable savings are possible. It's a very easy and pleasant trip. You may find it helpful to have a look at the excellent SEAT 61 website, as that has good information on how to buy tickets. Happy travels!
To get a discount fare on a high-speed daytime train, book ASAP (up to about 12 weeks in advance) at www.eastcoast.co.uk. To get a Bargain Berth on the Caledonian Sleeper, book ASAP (up to 12 weeks in advance). Discount fare tickets are for a specific departure date and time and are non-exchangeable and non-refundable. Be sure you can commit before booking.
Sorry to disagree with someone who lives in England, but Keith is wrong. While the standard 2nd class fare for London-Edinburgh is well over £100, advance discount fares can be as low as £17. I just checked the East Coast site and found fares as low as £28.90 for 14 March. (As of today booking is open through 15 March.) Bargain Berths on the Caledonian Sleeper start at £19.