George R. R. Martin likes to take names and titles from the real world and subtly change them, to give but a few as an example, Sir becomes Ser, Jeffrey becomes Joffrey, Melisandre is probably Melissa and Sandra (or Andre) combined, and so, presumably, Edward was used as the basis of the name Eddard. The books are full of such examples.
As Ned is an abbreviation of the name Edward, it is also, in Westeros, an abbreviation of Eddard.
As for why Ned is short for Edward, a quick google reveals little (except for a lot of Yahoo Answers noise), but here is, at least, a confirmation of the fact.