I think there are a couple of simple things that can help. As others have suggested, good lighting makes a big difference. There's several ways to do that, of course, depending on what effect you want to achieve. Hard lighting can be more dramatic and soft lighting can be more dreamy. Hard lighting is preferred in theatre and to an extent film. Soft lighting is more common in advertising and historically television. Multiple sources helps considerably. Ideally at least one is relatively close to the camera to provide fill. (I usually put mine slightly to the right and just a bit above the camera, but left can work as well.) A good overhead light can provide some striking shadows, but you don't want the shadows to completely overpower the details of your miniature. I have some pretty bright fluorescent shop lights above my gaming table and usually use them as the overhead. I'll add fill with some clip lights. I find both to be a happy medium on the soft/hard spectrum. Fluorescent tubes are pretty soft, but give good shadow below. The clip light next to the camera can warm the image up considerably and brighten the shadows up so you don't lose all the detail in them, but if the distance is right you won't lose them entirely. If you want a little softer fill you can bounce a bright light off a ceiling or wall. If you point it away from your subject and it's sufficiently bright the reflected light will be surprisingly soft. For a softer studio look without investing in a lot of equipment you can take a large sheet of stiff white or light colored paper bend it into a gentle curve. Place your subject on it in front of the curve. Use a bright light source reflected off the wall or ceiling, but not lighting the miniature directly. And again, use your smaller light source closely adjacent to the camera to fill in deep shadows.
I've found a typical smart phone camera works fairly well, since it usually has a lens with a relatively short focal length and a pretty wide field of view, and honestly, the resolution is pretty comparable to any but fairly high end SLRs these days. (Better than my own twenty year old SLR, though I still use the "real" camera when I want a longer lens.) It does help to have good software where you can change the focus and other settings manually. Depending on what phone you have there might be alternative software that's better than whatever came with it. And of course you can adjust color, crop, exposure, and add quite a few effects with software like Picasa or GIMP, both of which are free. (Picasa is out of support, but I believe you can still find it, and it's a very simple, quick and dirty editing tool. Very easy to learn and surprisingly powerful for how simple and fast it is.)
Anyway, it can be pretty rewarding. Play around. Have fun.