What is the cheapest possible way to travel from Edinburgh to London for a student with a very tight budget?

Ryan Air, or any other discount air carrier in the UK. Might cost just a few pounds for a student.

Ryan Air is the cheapest thing in that area. you WILL not get a cheaper price I promise you. I was stationed in england and traveled to scottland a a regular basis.

Megabus!

national coachlines

Megabus, National Express and budget airlines are the main options if not check liftshare.org or gumtree.com to share a ride with someone driving in the same direction.

Definitely Megabus. You can do it for under £10.

Ryan air is cheap but don't be fooled by the low price. The taxes and fees that are tacked on can be horrible. I would try train or bus.

Definitely Megabus but be prepared to travel in discomfort for long hours: http://www.megabus.com/uk/

Other options you can try are National Express (bus): http://nationalexpress.com/, National Rail (train): http://nationalrail.co.uk/ or Ryanair (plane): http://www.ryanair.com/.

The prices for all the above are not fixed so check out different timings and dates for best price.

And since you are a student, you can visit your School Travel Centre (if they are any) to check out the best prices. Also, you can apply for a Young Persons Railcard for 20 quids and it entitles you to rail discounts up to a third of the full fare for a year.