You have a digital input, an analog input and a digital output from your black box. I would use a microcontroller if for no other reason than to accommodate 'feature creep'. I would not waste my time trying to develop some analog setup using 555 timers and such.
So, given that Arduino is the easiest microcontroller to get started with and given that this system probably wants to work on 3.3V, I might pick the Arduino Pro Mini 328 and the required USB->Serial adapter.
https://www.adafruit.com/product/2377Once I had it working, I could dump the prepackaged board and design something that just used the chip. Or not, I have been known to mount these stamp boards right on top of a daughter board that handled or provided the peripherals.
Basically, your code would look for the input going true and, as quickly as possible, set the output. Now would be a good time to read the analog port and figure out how much time to add. Then continue to look for the input to go false and then start a count-down timer. When it times out, clear the output.
There is a millis() function for timing to milliseconds using an internal system timer or you can set up one of the hardware timers.
For the analog bit, you need to read the pot voltage and use some equation to convert volts to time.
I am sure there are analog solutions for this. I tend to focus on digital solutions and I'm really fond of the Arduinos, particularly for new users.