I passed a current through a solution of water and Epsom salt using a copper wire as an anode and a paper clip as a cathode. This produced a blue precipitate, which I filtered out. Can anyone tell me what this blue substance is?

 

 

Sounds to me like copper sulfate, though I could be entirely wrong on this as I am by no means an expert in this area.

Cu⁺⁺ has a nice blue colour like in azurite

http://en.wikipedia.org/wiki/Azurite

precipitates I saw have this colour but not as dark and intense.

 

Cu⁺ is green like in malachite

http://en.wikipedia.org/wiki/Malachite

 

Electrolyse salts but not dissolve the electrodes is difficult. Apart from gold, platinum and a few more... you can use graphite, which makes the central electrode of a banal zinc battery (the non-alkaline 1.5V elements). When opening an element, beware the liquid chemicals in it: protect your eyes, have water at hand. Better avoid it with an alkaline battery, it contains KOH.