What game will you remake? Why?
I would remake X-Com: Enemy Unknown. I would remake it because it's an amazing game which has seen sequels and spiritual successors all fail to do it justice, and I want one which DOES do it justice.

What would you do to build on the original?
I would keep things simple and streamlined. I would revamp the inventory system so that it's more intuitive and easy to use - perhaps you could assign loadouts to individuals, rather than craft, so they always have the gear you gave them. But there'd be an auto-equip feature as well, which would try to match available gear with people best suited to it. I would let you set filters so that you could choose better recruits without having to buy a bunch, sort through them, and kick out anyone who has low stats. This wouldn't be free, but it would be useful!

The game would still be divided into the three main parts of combat, geoscape, and management. Obviously, I'd pretty the game up plenty, but especially the geoscape. I would want that view to be a smooth, zoomable, hi-res globe that evokes The Blue Marble.

What would you do that would be significantly different?

There are two main areas of major change:

I'd revamp battles significantly. The core remains the same, which is to say turn based tactical combat. Beyond that, it'd be different. Modern technology would allow for greater battlefield conditions; rain, snow, etc. and these would have ramifications in combat. For instance, high wind would reduce accuracy and affect the move speed of flying units. Rain would reduce visibility. And so forth.

It would still be the same sort of grid it used to be, but aside from being larger, it'd be able to extend much higher. This wouldn't be necessary most of the time, but one of the new environments - CBD - would demand it.

Of course, new environments. There would be quite a few - wilderness, plains, subarctic, and numerous variations for inhabited areas. These would be; Central Business District, urban residential (Much as in the original), industrial, suburban, and small town. Now, this would have an effect beyond just the battlefield itself. It would also change who was there, because there would be a better-developed non-played Human element to the game.

Much, much better. Here's what I'm thinking:

Civilians would usually be present on all urban maps, except perhaps industrial types at night. However, they wouldn't necessarily be the panicked idiots in the first game. Some would flee (Trying to get into civilian vehicles to do so if possible), some would hide, some would try to greet the aliens (At least in the early stages, before the denials have to stop and everyone is fully informed of them), and some would try to fight back. Civilians would, depending on a couple of factors, have a chance of being armed with pistols, shotguns, and rifles. They might keep these until they're cornered, or they might take shots at the aliens from the start.

Factors would be gun laws in the region, and how well-known the alien presence is. Gun laws would probably be loosened as time passes and the alien threat grows greater, unless the aliens subvert the leadership.

This would mean civilians are a more active part of the battlefield; they mostly wouldn't run around in circles, they'd have objectives, and occasionally might even make themselves useful. But beyond the civilians would be police and army. Now, the police would be a moderately organized force, armed with the same kinds of things civilians can have, but also with a smidge of armor. They'd also be the first on the scene, arriving even before X-Com do. They'd seal the area off to prevent more civilians getting in, and their main objective would be trying to get civilians to safety.

The army would be slower to arrive but would, unsurprisingly, be a far tougher force than the popo. They would have decent armor, plenty of men, and serious firepower. They'd make the early game easier than it was in the original; the downside? They would quickly be forced to husband their forces, and be available in fewer areas (No more being deployed to Buttsmurf, Arizona to defend the 12 hicks and 244 cattle!), growing more and more reluctant to deploy outside of major cities and strategic locations. X-Com would be on its own for a majority of missions after mid-game.

The stages of the game: The early game would be before public knowledge of the aliens exists. X-Com is experimental and underfunded, but the aliens aren't doing too much yet. They don't go after targets larger than small towns. Not all governments are convinced about them, and some might not allow bases to be built yet. (Think about it; would Iran allow a multinational alliance composing large numbers of American, British, European, and Japanese forces to establish a very secret base in their heartland, on the basis of fighting aliens?)

At some point the aliens would consider themselves ready, and become more bold. This would mark the end of the early game. They wouldn't just launch themselves straight at Chicago, but their movements and attacks would grow more bold until they did reveal themselves in the first terror attack, and governments have to come clean. They start going after major cities to demoralize us, which is when the armies start getting pulled back and are much less available. On the plus side this convinces sceptical regions to join X-Com, meaning more money and more base building oppurtunities.

Governments would be capable of more, as you may have inferred. Some might try to make open pacts with the aliens; some might suffer under leaders who secretly make deals for their own protection. If one becomes extremely friend to X-Com they might let you give orders to their police and army units, if any are on the battlefield. Very good relations would also make them more forgiving of civilian and property damage, at least in isolated cases.

I could go on. But you get the picture. More weapons, more complex and larger battles, and so forth. But underlying it all must be exactly the same as always existed. An atmosphere of tension and fear. The feeling of facing a hopelessly superior force. Turn-based strategy. And Chrysalids.

tl;dr X-Com but better with cops and army and more guns and better graphics.