The simplest answer is a micro controller of some type. It would listen to the button and when pressed drive the display. To control your spotlight you would need other parts as your typical micro does not drive AC loads. The simplest micro could well be the very popular Arduino. Something like the UNO would be a good choice. Tones of information and sample code on the web. Free tools and not that expensive. You will need to learn C/C++ but that would be the case for any micro.
Lots of other micros exist including the MSP43 LaunchPad, the Raspberry Pie.
The other possibility is discreet logic, but that would be very messy.
As it turns out the display shield linked to has an ATmega328 micro on it. It could be reprogrammed to do the simple task required. A better choice would be this display, not in Arduino shield form factor, but having the same micro and w/the Uno bootloader preloaded. So some form of USB-TTL serial adapter would be needed but reprogramming becomes easier using the Arduino environment. And there are just enough I/O pins ,brought out to the edge, to connect to the dome switch and some relay/driver.
As for the C/C++ someone mentioned, that isn’t the problem for me. I’m a programmer by trade and have mechanical engineering experience and education, just not electrical - but I doubt the jump will be that difficult.
I’m just not familiar with the products on the site.
I’m planning on making more of these if it works, so If I can get a cheaper or more efficient board or parts that would be cool.
The LEDs are just for the prototype. I’m looking to swap the little LED for professional spotlights which I’m still trying to find a vendor for. Basically it needs to be bright enough to illuminate a person in red or green.