I've tracked down the behind-the-scenes story of Abbott and Costello's creation of the "Who's On First" routine and it's pretty fascinating. Evidently they'd worked up a rough version of the bit in their stage show and took it to their radio producers in hopes of putting it on the air. After an initial read-through produced fewer laughs than expected one of the radio people noted that the problem was that most listeners were already familiar with the Yankees' lineup, so after an all-night brainstorming session the duo returned with this retooled version:

Abbott: Well Costello, I'm going to New York with you. You know Bookie Harris, the Yankee's manager, gave me a job as coach for as long as you're on the team.
Costello: Look Abbott, if you're the coach, you must know all the players.
Abbott: I certainly do.
Costello: Well you know I've never met the guys. So you'll have to tell me their names, and then I'll know who's playing on the team.
Abbott: Okay, let's see, we have on the bags, Lou Gehrig's on first, Joe Gordon's on second, and Red Rolfe is on third.
Costello: Gehrig, Gordon, and Rolfe. Okay, that should be easy enough to remember.
Abbott: I would think so.

In the end, however, they found that this new script left 7 minutes of dead air at the end of the broadcast and thus opted to go with the original routine which we are vaguely familiar with today.