I thought about Congo Bill and Congorilla, but I thought Congorilla debuted after the JLA. Turns out that wasn't the case, but I think getting him in and out of Africa for every JLA adventure would have been awkward. Vigilante's a possibility, but he would really be out of his league. The Challengers already had their own team, as did Blackhawk and Rip Hunter (and the Sea Devils, but they debuted just after the JLA). Singling out one of them for JLA membership wouldn't be appropriate.
Rex would have been more helpful than Snapper Carr, though. And I wonder why they forgot Captain Comet in that gathering of contemporaries.

I had to go with Aquaman. I like Aquaman and all, but that one hour time limit where he has to go back to water or he'll die is so silly and arbitrary, and doesn't really make sense (it's always exactly one hour regardless of whether he's in a temperate rain forest or a desert) and is so incredibly limiting. I think Aquaman has a lot of potential as a solo character, but in a group of land based characters, fighting mostly land and air based villains, he's better off as a reserve member.
I'm not sure who I'd replace him with. Maybe Black Canary (gender diversity) or Cyborg (racial diversity plus some useful superpowers and a different background than the other).
I'm a little surprised to see Martian Manhunter easily "leading" this poll.

I'd replace the Earth-1 Superman with his Earth-2 counterpart.

You have to remember that there really weren't many options for DC when Schwartz and Fox brought back the JSA concept in the guise of the JLA.

Only Superman, Batman, Wonder Woman and the Flash even had their own titles when Brave and Bold 28 appeared in December 1959. Green Lantern had made just his third appearance in Showcase the month before and would not have his own title until May of 1960.

The only other superheroes doing business at DC in late '59 were Supergirl, Aquaman, Green Arrow and the Martian Manhunter.

So the only costumed heroes who weren't in the JLA in B and B 28 were Supergirl and Green Arrow, whose absence was an oversight, judging by his quick inclusion. Like Aquaman and MM, he'd never had a title and would last only another three years as a back-up. Supergirl was still a secret, so she couldn't join. (And Weisinger would have thrown a fit, of course.)

The other back-up characters were Roy Raymond, Tommy Tomorrow, and Congo Bill. Roy was a TV detective, Tommy lived in the 21st century -- how futuristic! -- and Congo Bill was in Africa. And all three would soon be making their final appearances anyway.

The other DC titles featured adventure teams, science-fiction, humor, romance, war, and mystery. No costumed heroes need apply.

I always thought that Fox wanted Adam Strange in, as he was mentioned as early as JLA 4, the issue when GA was allowed in, as a possible new member, along with Hawkman, who was in just his second issue of B and B that month. But Adam was also not really a super-hero and would have preferred domestic life on Rann with Alanna, as any sane spaceman would have.

With the revival of interest in superheroes, JLA tended to respond pretty quickly when the remaining Silver Age counterparts received their own titles, with new members in issues 9 (Atom, who was only in his second issue) and 31 (Hawkman, after he finally had taken wing in his own comic seven months and four issues prior.)

I voted Batman for a variety of reasons, mainly because I think being a member of such a national, if not global, team works against his very mantra. If your schtick is to inspire fear and dread in your cowardly criminals through subterfuge and shadows, how well is that going to hold up in what I imagine would be the very highly publicized exploits of the JLA?

Plus, in recent decades, I hate how it's become so common and even nonchalant over how Bruce is now smart enough to take down any and all foes from lowly street punk to cosmic Darkseid (including his own teammates).  It pushes to the breaking point my suspension of disbelief over this mere mortal man who dresses up in a batsuit to fight crime.

Instead, I'd rather see Batman more as a support member, perhaps brought aboard in certain situations and crisis more suitable to his capabilities and MO style.