This can be a pretty simple circuit.
1 - CD4060 oscillator/counter to set the blink rate, count the number of blinks, and stop at the end of the cycle.
1 - transistor to drive the indicator light
This circuit works best for a fixed number of blinks per cycle. A 2-chip circuit (LMC555 + CD4017 + transistor) can be built to be adjustable from 1 to 9 blinks by moving a removable jumper.
What is the indicator light? LED? small incandescent? Voltage? Current?
At the end of the cycle, does the light remain on or remain off until the next button press?
ak