Absolute minimum features? To be honest, I'd be overjoyed with a remake of JA2, just with better graphics, in a new setting.
That said, top of the list is 'turn-based'. I can't even bring myself to Google the names of the real-time pause efforts of recent past - it just has to be turned based combat, end of story.
Moving down the wishlist, yeah, I think the merc personalities should be a thing. Humour, and off-beat humour at that, should definitely stay. Building on the rivalry thing, in JA2, there were one or two rivalries you had to watch out for, otherwise one of them quit. This mechanism could stay, but be expanded upon - quitting doesn't have to be the final result. Other nerfs could be implemented to reflect inter-merc disagreements.
Stealth mechanics in JA2 were ok, but could really be improved upon. Somehow related to this, I want to say something about martial arts - kind of implemented in JA2, but you could never use it - it was just weak. Some kind of character build with stealth/martial arts would be superb. There would need, though, to be a mechanism to prevent both stealth and martial arts being spammed. Or maybe not - I dunno.
Next up, neither JA or JA2 really got the roster right, great as some of the characters were. In both games, a large proprotion of the availble mercs were useless at any point of the game, and nobody ever hired them, unless they really wanted a challenge. Put it this way - if you're hiring Pops McGillicutty because that's all you could afford, you might as well start a new game. The lower third of the entire roster in JA was junk, and much of the mid tier. I'm looking at you, Tex R. Colburn, Jimmy Upton, and Wink E. Dickerson - just no use, at all. In JA2, you at least had the likes of Bull, the cheapest Merc in the opening game, but at least he could open stuff for you - and he often survived to just do that, right to the end. Realistically balancing the economy with merc abilities should be key, with far less 'filler' mercs. Opening the game with Ice, Ivan, Fidel, and user-defined merc gets boring.
Moving on - maybe a bit of randomisation of stats among the well known characters. One of the things that grew tiring, as replayable as JA2 was, was that everyone was exactly the same, every time. It would be interesting if stats were changed up each game, meaning that when you're looking at the AIM roster at the start, you really do have to look and evaluate each time. Reviewing each character's value (both practically and financially) would breathe new life into things.
Stand/crouch/prone - say no more.
Weapon customisation - yes.
Strategy layer/tactical layer - yes.
Free roam on the strategy layer, as opposition permits - yes.
Resource procurement from captured sectors - yes.
Fragments of mystery notes the player needs to figure out within a time limit (like JA) - hell yes.
Voice-over descripton of objects with inappropriate amounts of reverb; 'A sausage - in good condition!'
There's much, much more I could list as things I'd like as a minimum, but much of the above is really a wish list. If we ever even see this game, I'll be amazed.