LED Blinking with 555 on button press
This can be done with a 556 timer IC (dual 555 in a single package), or two 555 ICs.

Use the first "555" of the 556 (or an individual 555) in a bistable multivibrator configuration, as described in simple terms in this article and this other one.



Instead of using the output to drive the LED as shown, feed it to the RST pin of the second 555, which would normally be connected to the supply rail. Configure the second 555 as an astable multivibrator to blink your LED:



Each time the pushbutton is pressed, the output pin of the first multivibrator will toggle state, and the blinking output of the second multivibrator will thus start, or stop. No microcontrollers are needed.

I'm too lazy to draw a combined schematic, but the above should serve the purpose.