The mechanism is likely a combination of a linear polarizer and a circular polarizer (which itself is a linear polarizer followed by a quarter wave plate). Thus the differences you can expect between high-quality and cheaper variable ND filters ought to be similar to the differences found among polarizers which include flare, vignetting, inhomogeneity, and color shifts. You can avoid flare in many cases by not including bright lights in the photo; vignetting can be compensated in software and sometimes is desirable; I suspect the inhomogeneities are usually not noticeable; but color shifts can be pronounced. Typically a set of crossed polarizers (which is what you're using to get the heaviest ND settings) pass essentially no colors except violet. Therefore, I would expect cheaper variable ND filters to work OK except at the heaviest settings (more than about 4 stops) where the violet shift will become pronounced.

You can see evidence of vignetting and a violet shift on this dealer's website. Compare the bottom two photos: the one taken with their ND filter has pronounced vignetting (many stops) and a strong blue cast. In contrast, look at the photos on the Singh-Ray site. If you look closely you can see the vignetting, but there's little evidence of a violet shift: look at the grays of the rocks in the stream. Maybe those were post-processed to remove the blue, so the comparison is not definitive, but these two sites nicely illustrate the difference we would expect between good and bad variable ND filters.

I did a quick test using a high-end, top-rated Marumi circular polarizing filter and an old low-end Vivitar linear filter. In combination they made a fine variable ND filter, although there were some color shifts (first to yellow, surprisingly), but below about 3-4 stops a yellow incandescent light started to take a distinctly blue hue. At the maximum density, probably around 6 - 8 stops (I only looked, I didn't measure), only the light was visible and it was a brilliant blue, the color of a Wratten 80b filter.

Your best bet for a cheap solution, then, is actually to look through one of the inexpensive filters and pay special attention to the apparent color at the densest ND settings. While you're at it, look at a bright light through the filter to check for flare and scattering. If everything's acceptable you probably have a great deal.