The JLA is doing pretty well under Meltzer, overall, so I'm mostly cool with the JLA we have.
I'm generally not fond of messing too much with what works, so conceptually I'd keep the current direction. The JLA should be the greatest heroes in the DCU, so the Big Three at least should have a place within the League. At the same time, I tend to share the view that there need to be a class of second-tier characters that can change and grow. Likewise, I would want a mixture of big action and characterization.
What I would change would be more the details than the foundation itself. I'd change the lineup somewhat, though probably not too significantly. I'd want to focus on the relationships between the members of the League, as that's the one thing you can find in the JLA that's not so prevalent with some of the other books.
The villains also need to be the worst in the DCU, to pose a challenge to the JLA. A new Legion of Doom led by Luthor would be where I would start, and I'd look for some of the most dangerous villains in the DCU. (And yeah, I'd go with the Legion... the Injustice Gang lacks to me as a name.) I'd also want to pit the JLA against some of the serious nasties developed in 52... Black Adam, the Oolong Island scientists (led by Morrow and Sivana, with a rebuilt Four Horsemen), Mr. Mind, Lady Styx, and so on. I would like to see the JLA take on the cosmic heavies... Darkseid and the Sinestro Corps especially. The JLA can only be the best of the best if they fight the worst of the worst.
My lineup is by no means definitive, but here's who I'd generally look to...
Superman: The heart and the moral center of the Justice League. The guy who always knows what the right thing is, and inspires the rest of the League to be better people and better heroes.
Batman: I would use the current Batman appearing in the solo book. The Batman that is human, doesn't always win, and struggles to act like a human being in spite of himself. He's a great conflict catalyst and good at tackling the mysteries that the team would have to unravel. No BatGod stuff, though... I never want to see that portrayal ever again.
Wonder Woman: She would need to be revamped, but that's a job for the solo book. ;D In the context of the League, I would say that Diana is there as part of her mission of peace, and sees herself as a peacekeeper who understands war too well to ever choose to wage one. She would be a teacher and an advocate, and also the ideal person to train the recruits.
Martian Manhunter: I always felt it was a mistake to take him out of the relaunch, due to his history with the League and the potential of the character. J'onn J'onzz is the mysterious solitary alien figure who yet chooses to stay with nearly every version of the League. I'd always felt that J'onn uses the League as a means of keeping human contact... but I see his reasons being more involved in that. As a telepathic entity, what if he keeps close to the heroes to keep himself grounded and sane? This guy survived the destruction of his entire planet, including his family... he's lost everything in his life. And his sanity would be in particular question after what he endured in World War 3. Lots of options with him, I think...
Green Lantern: I'd go with Hal Jordan here, and admittedly he's a favorite of mine. That said, he brings some good dynamics and adds an element of impulsiveness and reckless adventurism. I like the contrast between Hal and Batman particularly here... the (mostly) fearless and impulsive space cop against the fearsome and calculating dark vigilante. His arrogance and strong will gives a good spark of tension, which I rather like.
Flash: Wally West. I like him because he gives a grounded "normal" perspective to a team full of icons and massive egos. He's also the bridge between the elder Leaguers and the younger set... the guy who began as a sidekick and grew into his mentor's role.
Atom. Ryan Choi. Because I like the idea of a new legacy hero trying to fill the shoes of an icon, and the new Atom is quite interesting. He should represent the reader viewpoint quite well too. I also like the balance between Atom's scientific intelligence and his deep insecurity... and there's nothing as unsettling as taking your idol's place with the biggest heroes in the DCU.
Zatanna: I'd like to see Zatanna, in her Morrison portrayal, play off against the rest of the League, especially after her struggles in Seven Soldiers. I admit that I'd like to see Zee play off against Atom, too... the naive rational insecure newbie against the more confident experienced sorceress with serious issues... they'd contrast quite nicely.
Hawkman: Another conflict catalyst choice, especially in the clash between his violent brand of justice against the likes of GL, Superman and Batman. Because every team needs its own barbarian warrior to stir things up.
That's where I'd start with, at the least. I'd like some less conventional choices, too, but there aren't many names that jump immediately to mind that would fit too well. I'll think on it some more.