Here is one theory put forth by Botley, (who I would believe pretty much anything he said concerning the Floyd):

"I like the idea that "Eugene" was a roadie carrying a plugged-in guitar which happened to be connected to all sorts of funky fx pedals while the Floyd were practicing a new song. Roger was mumbling random whispers into a mike when he saw the roadie walk past. He mumbled "careful with that axe, Steve" (insert the name of a real roadie). Steve (or whoever) was so surprised to hear his own name in the new song that he dropped the guitar right on the strings, making an ungodly howl that Roger promptly imitated with his patented Scream. The Floyd recorded the song a few months later, after the roadie had been fired for dropping equipment, so they had to change the name. The most convenient name available happened to be one they just made up for the A-Side of Axe: "Hey Eugene this is Henry McClean..."

They never bothered to explain this to Storm Thorgerson, of course, who assumed that they meant a real Axe, so he stuck one in the Echoes artwork."

The above was from the following thread:

http://www.pinkfloydfan.net/showthread.php?postid=38115

My guess is that it was a name picked out of thin air by Rog cause it sounded amusing.

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