Actually the number is not as random as you think, but still has a very strange story to it. I think I read on Vintange Mustang Forums that the name came to be when someone asked how many steps it was from one building to another. Carroll Shelby said "about 350" & thats how the name came to be.
I have the Motor Trend tribute to Carroll Shelby magazine and it has a small insert from Carroll Shelby about the origin of the name. Here is what Mr. Shelby said:
I remember when Iacocca wanted us to build the GT350. We had five or six meetings. A planeload of people came from Detroit trying to decide what to call it. And after about the third meeting, I said "If Nissan can have Ladybird as the name of their sport car, a name doesn't make a car. A car makes a name."
So we were all sitting around there (Shelby's facilities at Los Angeles International Airport), and I said to one of our guys, "How far is it to that building over there?"
And he said, "What the hell are you talking about?"
And I said, "Step it off."
He came back and said, "It's 348 paces."
I said, "We're going to call it the GT350, and you guys can go back to Detroit."
RIP Carroll Shelby