I don't know the physics of skipping in detail, but I can make a guess based on my knowledge of fluid mechanics. You probably know that water (and other fluids) has a "surface tension." Basically, the surface tension means that it costs energy to create new surface area. This fact means that the water likes to minimize its surface area. A consequence of this is that some objects that are denser than water (and would ordinarily sink) will rest on top of the surface of the water if they are placed there carefully. By placing the object carefully on the surface without disturbing the surface too much, the object must overcome an energy barrier in order to sink. The origin of this barrier is the need to create additional water surface area.
So, back to skipping. If an object impacts upon the surface of water with a small enough vertical velocity, then it may be the case that it does not have the energy (momentum) needed to break through the surface of the water (overcome the surface tension). The surface may act like a spring as the object hits, and eject it back into the air. A flat object would work best because the force of impact would be spread over the largest area possible. I think that the larger the contact area of the impacting object, the more surface area must be created for the object to pierce the surface,
and thus the easier it is to skip.
Lastly, I think that the horizontal velocity plays some role also. Namely, I think that if the object is moving too slowly horizontally, the friction during impact will cause the object to "stick." That is it will come to rest and will sink.
I can't find anything about skipping in the books on my shelf, but you have made me interested in this topic, and I will see if I can find anything elsewhere. I will give a warning to your student that I suspect that this is a very complicated fluid mechanics problem (I could be wrong). The theory is probably very difficult, but experiments can probably be done.
The important variables would of course be object shape, object size and density, (possibly) object surface roughness, object velocities (horizontal and vertical), and the surface tension of the water or fluid being used.

Skipping stones is a tough one.This is closely related to the 'slamming' problem that probably first arose with seaplanes during WWII and has continued in understanding slamming associated with ships in heavy seas when the bow may come out of the water altogether ad then slam back down. The are conditions for which most ships are not designed so there has been a lot of interest, but not so many solutions.
Large forces are exerted by the water on the body seeking to enter it. Basically the density of the fluid and the density of the body are not so different so it is not unlike two billiard balls colliding. In the case of the skipping stone it approaches the water at a low grazing angle and so it rebounds much like two billiard balls. If the angle is too low though, the viscous losses associated with the fluid resisting the motion will probably slow it down enough that it may not escape the surface. If the angle is too high the fluid swallows the object; there is too much enrgy loss in the inelastic nature of the 'collision'. Much of the kinetic energy is lost to dissipation and the object can not rebound.
One of our former graduate students worked a great deal on this problem and has written several technical papers on this problem, but these are highly mathematical papers that seek to solve for the fluid motion around the object entering the water surface. In this case, if my memory serves me,he looked at a cylinder slamming into a fluin at rest. There are many subtlties associated with this problem especially related to the point where, water, solid and gas all meet. This point migrates. The bottom line is that very large faorces arise due to the need for fluid to move out of the way of the object, these forces are of short duration and difficult to measure and/or model.