Well, I don't really like the other suggestions, as they aren't really demonstrating anything, they are just explaining things. But I do have an example of a real quantum phenomena that can be performed with a light source (preferably a laser pointer) and some polarizing optics. Polarizing optics can be polarizing filters for a camera or the lenses from a pair of polarized glasses (you will need three polarizers, which would mean 1.5 pairs of glasses). If your light source puts out polarized light, you can get away with two polarizers. And in a pinch, you can use a piece of glass oriented at brewester's angle for the first polarizer.
But here's the gist of the demo. Shine the light source through the first polarizer so that you make a beam of polarized light (again, a laser pointer should already be polarized so you can skip this step if your laser pointer is working correctly).
Take the second polarizer and demonstrate that the light is polarized by shining the polarized beam through that polarizer and rotate the polarizer. At some angle all the light should pass through the second polarizer. If you rotate the polarizer by 90 degrees, none of the light will pass. This demonstrates that the first polarizer makes vertically polarized light and the second polarizer blocks vertically polarized light (passes horizontally polarized light).
Hold the two polarizers so that none of the light gets through. Take the third polarizer and place it in between the first two. If the second polarizer is oriented the same way as the first polarizer, it will pass vertically polarized light, so the light will pass through the first two polarizers but be blocked by the third. If you rotate it to be oriented the same way as the second polarizer, it will pass horizontally polarized light so the light will pass through the first polarizer but get blocked by the middle polarizer and never make it to the last polarizer.
But here's the quantum. Take the middle polarizer and rotate it to be 45 degrees from the other two. It will now pass horizontally polarized light, which is a superposition of vertical and horizontal light. Now the light will go through the first polarizer, half of it will go through the middle polarizer, and half again will pass through the final polarizer. So at this orientation of the middle polarizer, 1/4 of the light will pass through all the polarizers.
Even though the last polarizer blocks all light coming from the first polarizer, the middle polarizer did something to the light that aloowed the light to pass through the last polarizer. This is actually very fundamental quantum mechanics and demonstrates some powerful ideas of quantum using simple inexpensive devices. It demonstrates superposition and measurement phenomena, and you can even make an argument that it demonstrates uncertainty in some ways. There are much more expensive experiments done in research labs attempting to demonstrate this very effect in other systems, so this really is fundamental quantum mechanics, on the cheap.
If you have a pair of those new 3-D movies glasses, that can be even more fun, because they act as circular polarizers (if orieneted correctly) which can be fun to try to figure out what they are doing.
In the absence of a laser or a bright light, you can also do this demo using a polarized display. Some LCD devices (computer monitors, cell phone screens, etc) have polarized displays. The disadvantage to this over a laser pointer is that they aren't bright enough to see a projected beam, so you have to do the experiment looking directly at the monitor, which means it only works for the person holding the polarizers, and does not work well as a demo.
I'm a quantum physicist doing more expensive quantum research.