The cheapest way I think would be Megabus. But you would have to go via London. It would take a long time but could be as cheap as �4.50 return if you book in enough advance.
You'd have to book 2 separate tickets. (or 4 for a return journey). Megabus goes from London to Brighton so it should work. If you are returning to Birmingham you'd have to book a return from Birmingham to London and also a return from London to Brighton.