You can build a delay very simple with a 555, with half a 4093, or with a couple of transistors + glue.
To go with the first two, they could be trig'd from either an electret microphone capsule and a transistor, or with a piezo disc if you only want it to trigger by loud sounds like clapping your hands (or by tapping near the disc ).
What output do you need when triggered, before that and after the timed part??, something like this: __|¨¨¨|__ ?
What supply voltage?