Trigger Signal Lockout Circuit
I would say: Stage 1: monostable 555 circuit then take the output of that into Stage 2: invert the monostable output and "and" it with the 555 output (DeMorgan to simplify as needed). Then if you want to control the output pulse width you can use the other half of a 556 to make a "one shot" of the desired pulse width.

There are some ill-defined details to work out about leading/falling edges and so forth but the above architecture should serve as a good starting point.