I haven't gotten through all the posts, so any duplicate suggestions may or may not be intentional. Some will be for added emphasis, like a better AI, which seems to be the top request of many posters.
Having said that, my wish list (and this is exactly that, I'm not going to get too concerned with practical considerations):
1. A Better AI. If there is no other improvement, then this is a must. Most of the other issues I have with HOI 2 are minor, especially compared with HOI 1 (except for the tech tree, to an extent. More on that later.).
2. Air Transportable infantry units. Strangely enough, transport planes functioned perfectly well long before people got the idea of throwing other people out of perfectly operational aircraft. It'll require a bit of research to determine exactly what type of units qualify - Armor/mech/mot are right out, obviously, but most infantry type units should be transportable. Just not droppable (with any expectation of survival, at least.)
Sorta like Franco's army to Seville, except not abstracted, even if it requires more than one trip...
3. Kiel Canal. I was shocked,

shocked, I tell you, to find that vanilla HOI 1 and 2 both neglected to include it. Kudos to CORE for including it in their 2 mod.
4. A much more abstract, complex and intelligently designed espionage system.
No sensible co-intel operation kills out of hand any spies it finds. More typically, the spy network would be surveilled with the intention of rooting out the entire network, attempting to turn or interrogate operatives or feeding them dis-information. If all else fails, then you do a big show trial to embarrass their masters and
THEN you shoot them. In game terms, a big relations hit here.
A better system would be one that is fairly hands-off. You set a budget for the target nation(s) and let your agents (they hate being called spies - that's the other guys!) do their thing. Success or failure would be based on several factors such as relative technical expertise differences (think counterfeiting passports, spy cameras, radios, etc.), types of governments (open societies being generally easier to penetrate than closed), mission types, relevant ministers, dissent, funding, etc...
Nations seperated by water or by other countries should be naturally harder to penetrate, and very much more so when at war. Typically, many embassy employees would be actually intel operatives with diplomatic cover (and immunities) and you would lose this option once hostilities ensue.
Failures might entail reduced effectiveness of some or all spying in that country, or even in the country's allies, coughing up good intel, or to the extreme worst case scenario of having a spy turn double and actually disclosing YOUR secrets to them, all the while still working or pretending to work for you.
Successful ops may provide tech blueprints (the postwar KGB and GRU excelled at this), partisan group formation/support, sabotage ops, research interference, etc.
Military intel in the field should be a separate process, but possibly could be supplemented by espionage - troop concentrations, possible invasion/ defensive preparations, new unit types, etc...
There's really a lot of fun possibilities with this if done well. Spying is really rife with broad gray areas where you're just not ever sure about things - even if you have a government that likes to consider
all the intel. (Ahem...)
Sorry if this is long, it just really stuck in my craw from the second I read the espionage section in the manual and I've been thinking about it a lot. That and I'm a big espionage fan.
5. TC should be divorced entirely from IC and depend entirely on infrastructure. After all, what
is Transport Capacity but the ability of the infrastructure to move units and supplies? This would make infrastructure actually significant in the game. As is now, it's almost of negligible consideration. And possibly, IC should actually detract from TC. Think goods and workers needing transport to the factories and all.
6. Shipyards. I believe this has been covered fairly sufficiently, so this is just my seconding the motion.
7. I also agree with the idea of some specialization of industry, with the reminder that factories could be and were re-tooled. GM here in the states produced tanks instead of cars during the war. It took a bit of time to re-tool, but not too much since it was already in the bidness (yeah, that's a Texian word) of vehicle manufacture.
8. U-boats. By their very nature, subs are meant to be fairly independent. Too many in the same area could end up killing each other, especially without wolfpack tactics or good radio comm gear (and that doesn't work at all well underwater).
But even so, large groups of subs in a small area is a bad thing. I'd love to see Prien's penetration of Scapa Flow to sink the Royal Oak emulated. Subs were very powerful, and yet very vulnerable and I don't think the game currently captures this aspect well.
9. A much better command/unit structure. I find it very odd that a Major General commands a division only as long as it's operating independently. As soon as it's incorporated into a corps or larger, suddenly it loses its commander. Very unusual. I'd like to see the traditional Division/Corp/Army/Army Group structure implemented, while keeping unit commanders who can add their traits to battles for that unit. Sort of a cumulative effect, but not always one that adds. For instance, Guderian during the invasion of France wanted to exploit his breakthrough but was held back by his higher command.
Or the idea that some generals were more suited to a higher command than a lower, and vice versa. Patton would never have done as well as Eisenhower as Supreme Allied Commander. I realize this might be very difficult to work out, though, but like I said, wish list.
And especially, working out a way to issue orders to just the Army Group HQ's and letting them sort out the details, unless you really, really want to be like Hitler and tell everybody what to do, where to do it, how to do it and how long to do it.
This gets back to that improved AI...
10 Food production in provinces. Scorched Earth Tactics. Improved resource production. (New oil well drilling, mining, and the like.)
11. Arms sales to minors. I know the Turks bought one German battlecruiser, the Yavuz Sultan Selim (ex-German Navy Goeben). Although this was in the '20's, they did buy some German subs and Italian destroyers in the 30's as well. CORE has plane sales to minors, but not naval. Both should be included.
12. Tactical air in very specific provinces. I know this is already been mentioned, so this is merely a 'Me too!'. Counter air missions, someone's already mentioned this as well, and the ability to destroy a/c on the ground. Since this really was a major German tactic, I'm surprised that it hasn't been included in the previous HOI's.
13. Rail guns. I dunno why, I just like the idea. They existed, and they're cool. But they do bring to mind the problem of the different rail gauges used in Russia and Germany that precluded their employment in the east until the tracks were re-laid to accomodate the German rolling stock. Heh. That'll be a bit of a nut to crack in game terms. Probably not cool enough to justify the work by themselves.
14. The SAS, Devil's brigade, Rangers, and all the other special forces used. I mean, we got commando commanders, already, right?
15. R & D should be based largely on money, I think, and relatively less on IC. The Manhattan project worked out for the US mainly because we threw so much money at it. Even though it did require building quite a few facilities, these were out of the norm for R & D work and even then really didn't require that large of an IC commitment, compared to say, building a carrier or an armored div. It was having enough brainiacs in one place that was the essential element. So maybe manpower should have some effect? That and the educational system, which might be hard to put into play. But still, the IC required should probably vary as well as cost, depending on the tech involved. Also, in relation to espionage, why not have stolen (or traded) blueprints give a variable benefit instead of 50% always? I'd almost think traded techs should be 100% anyway, but I guess that could be negotiable...
I kinda liked HOI 1's tech tree scheme a little more than 2, but it was way hard to figure out the prerequisites for the techs if they affected other fields. So maybe 1's tech sheme with some nice pop-ups with info or something?
16. Re-do the victory points for the US. Really. Much as it pains me to say it, New York really oughta be worth a lot more than New Mexico or Utah or Montana or Alabama. Most of the points should be in NY, Washington DC, LA (Horrywood! Heh, sorry. 1942 - John Belushi movie...), Seattle (Boeing), St. Louis, Fort Worth (Vought), Houston (awl), Massachusetts/Connecticut, Norfolk, Va., Detroit... (all major population/industrial/military production centers). Not meant to be a real comprehensive or even accurate list, just off the top of the head.
17. And lastly, could we get progress bars or some indication of work done on units in production?
I think I could think of more, but my brain hurts so I'll stop. For now, at least.
D'oh! I knew I'd remember one more thing just when I thought I was done.
Sliders. I hate having to wait on the various production sliders. MOO3 has a nifty feature I'd like to see in HOI. Click once = 1 point. Shift + click = 10 points. Alt + click = 100 points. Something like that.

And also maybe being able to just type in the number you want in addition to mousing the sliders?
R.