What about lens construction influences bokeh?

I can understand how certain elements of a lens can correct an aberrations or make it sharper - but specifically, how do lens manufacturers influence the bokeh - the quality of the blur? There's a huge difference between the bokeh of say a 85mm f/1.4 and the standard 18-55mm kit lens.

Answer

As has already been pointed out, out of focus highlights will (mostly) reflect the shape of the aperture. That may be the most obvious element of bokeh.

There is quite a bit more to bokeh than just that though. Spherical aberration correction is another important element. If a lens corrects spherical aberration perfectly, an out of focus highlight will simply increase in size compared to one that's perfectly focused, but the light will be distributed perfectly evenly throughout that disc. In reality, however, that isn't how all lenses work, and (in particular) it doesn't lead to what many think of as the nicest bokeh.

In reality, the illumination across that disc is often at least somewhat uneven. What most think of as pleasant bokeh results when the center of the disc is bright, and the brightness falls off toward the edges of the disc.

The opposite is also possible though: most of the light falls toward the edges of the disc, and the center is relatively dark. In severe cases, this leads to "double line" bokeh -- the center part of a disc is enough darker than the edges that the center seems to disappear, and what was one object looks almost like two instead. Even in less extreme cases, this tends to lead to a "nervous" looking bokeh where features and textures tend to stand out rather than blending together nicely.

The third possibility is that even though the same overall brightness is maintained across the disc, it's not the same color throughout. One typical manifestation of this is discs that appear to have a green outline around the outside.

One more point to bear in mind is that exactly the same lens will often exhibit more than one of these. In fact, nearly all lenses exhibit all three, to some degree or other. The first two (bright center vs. bright edges) will depend on the position of the out of focus highlight relative to the focus point. If the background shows bright edges, then the foreground will typically show bright centers (and vice versa). Nearly all show at least some minimal degree of color change across the disc, though it's certainly much more visible in some cases than others.