Unless they have changed the design...
When someone places their weight on the square, it sinks down, and the foil touches the Positive and Negative strips, causing the circuit to complete.
While they Might have also used the pushbutton as a backup... I think they choose foil as the main contact method, because microswitches need a 2mm depth to activate.. and often need perfectly centered activation. In the case of a Dance game.. players might often barely tap a tile quickly, rather than completely put all their mass on it. This is probably especially true, for the harder levels, whos notes fly by so fast as to nearly break the sound barrier.
The originals probably had a Foil pattern on them that was shaped in a certain way to allow the light to shine around it. Such as possibly leaving a cut out where the arrows white outlines are. But as they wore out... and operator probably just put more tape in other spots, thus covering up the lighting.
If you have a way to get cheap prints, you could probably scan the arrows, touch them up.. print them out. Use paint remover on the plexi, sand and polish.. then apply the printed arrows to the bottom of them.
You will then have to get some sort of metal or foil contact material, and glue it to the bottom... leaving the areas you want lit up, uncovered.
Not sure how much new arrows go for... so not sure which method is the Thriftiest.
The xmas lights do not work too well in this application however. They are too focused, and the light does not seem to spread out enough... thus causing them to be quite visible when lit. This is why Pinball machines look best with Traditional bulbs, rather laser beam retina scorching LEDs. A few pinball lamp sockets would probably be much better... and look a lot better. It would be interesting to put some pinball flashers in there for fun. heh. Would probably give quite the startle, as the player may think an electrical short just happened! heh
You might be able to salvage the xmas lights... by balling them up, and covering them with a translucent, but not transparent, shield material. Sort of like how certain Pinball machines use those rubber covers over bulbs. Or a cars brake lights have an internal diffusion surface inside of them.
xmas lights are pretty unreliable (even the LED versions), and usually dont last more than a handful of hours.. which would be a real pain to deal with regularly. Especially if they are out of season. Where as standard small socket pinball bulbs can last years of constant "On time", without failure.
Better quality LED can last a lot longer, but there is added complexity, costs, and the focused lighting issues... as well as possible color issues. Its only recently that they have created white leds that do not have a blueish tint. Many of the xmas sets dont have them however, as they are probably trying to get rid of the older leds. Also, not sure of the lifespan on new white leds. Certain solutions get the desired end results... but not the longevity.
Also, for better light spread... paint the inside of the box white. Such a thing can reduce the need for brighter bulbs, and produce far superior results altogether. Dont be tempted by metalics / silver. White works better. I speak from experience, when trying it out on a tube fixture.