I use the Axis Blade, because it's just awesome looking, and it's pretty nice to get a constant stream of attacks because as long as you synthesize the ATB charge (by using Sprint Shoes, Aurora Scarf, Whistlewind Scarf, among other things all at once), Lightning with the Axis Blade makes a perfect juggler all by her lonesome, which leaves the other two to do other things without having to get the timing down. Granted, you will then have to use Lightning as your main just to be sure (as to juggle in that state with 20% ATB, which is reasonable, you have to triangle-cancel before the last attack sometimes to match up animations to keep it steady).
That said, the Axis Blade is seriously weak compared to some of the others. REALLY WEAK. And the nice thing about Lightning is that she's an offensive powerhouse because she has powerful magic and strength, unlike every other character, which usually has a particular style/role. So yeah, I dunno. Eventually, you'll be rolling in money, and upgrading weapons to their max in the first tier is going to be pocket change. So I would just go for it, and do it with some of the other weapons, and see what you like.