What to consider when changing V-brake for a disc brake?
First, it's an expensive upgrade that won't benefit you much. You'll see quite a few cheaper bikes that only have a disk on the front, because that's where you get most benefit from a disk. So the cheaper option is definitely to buy a better V brake. It will work just as well and be cheaper.

You will need a new rear hub, and it's likely to be cheaper to buy a prebuilt wheel rather than pay someone to rebuild your current rim with a new hub into a wheel (I suggest not re-using the spokes even if you can). You will, of course, need to buy the disk brake as well.

Since it's the rear brake, I'd go with the cheapest cable operated brake I could find. The advantage there is that it's cheap. A more expensive cable operated brake will work slightly better and have both pads adjustable making it easier to set up. But it won't look any more like a disk brake, so there's no real benefit in your situation. Likewise, a hydraulic brake will work better in the rain and snow, but cost more.