How to build a programmable sticky button

Someone has already invented this.

The circuit breakers used to simulate aircraft breakers in flight simulators can be released by a small current, but they are not cheap.

The device pictured sells for hundreds of dollars. I see a relay mechanism used like a firearm sear, some contacts, and a spring-loaded plunger - you may be able to duplicate this at a lower cost

There are solenoids that can pull in instead of push out. Ones with springs on them so that they default to out.

Taking one of these, and you can build a simple latch. When the button is pressed down, the solenoid arm is pushed into a hole on the side of the button, locking it in place. You can then just pulse the solenoid to unlatch and the button pops back up. So no need to power it all the time. For this you would need to fabricate a button cover and spring setup (hello 3d printers). Or you could modify an arcade button type switch with a drill. How you do it would depend on the solenoid you use.

Alternatively you could make it electronic. Instead of keeping the switch pressed in physically, have it light up an led. You could get button switches with LEDs built in that you can control. This would mean you need to power it, but it will be much less current than a solenoid.

Maybe a small servo could be used to pull down the push button stem.