The simplest solution is to use a CMOS Schmitt Trigger such as the 74C14, 40106, etc. Connect a resistor to the input, and a capacitor from the input to gnd.
This will delay the signal applied to the resistor (ie. a step function) by about one time constant, ie. R * C
It won't be very accurate, but it is cheap. You will have to adjust R and/or C on test to obtain the delay you want.
Also, the "off" delay will be about the same length. To reduce the "off" delay you have to speed up the discharge of the capacitor. One way to do this is to connect a resistor in series with a diode and connect them in parallel with the resistor.
Len