Below is the LTspice simulation of a circuit, to do the pulse detection and stretching.
This is the simplest analog circuit I could generate.
I first tried using 555's but that became even more complicated.
This circuit uses one LM339 comparator chip, one CD4013 flip-flop, two diodes, and some passive components.
The two LM339's, U1 & U2, are configured to provide about a 5 second delay each.
U4 is configured as an input buffer.
The simulation is shown for an input pulse of >5s.
≈5s after the input pulse V(input) goes high, U1 Out goes high.
This triggers the flip-flop, U3, causing its Q Output to go high.
At the end of the input pulse, U2 starts to time, and after ≈5s its output, Out2 goes high, resetting the flip-flop and terminating the Output.
The circuit thus gives an output pulse that starts when the input pulse exceeds the 5s delay, and continues the output pulse until 5s after the end of the input pulse (hysteresis?), which I think is what you wanted.
These periods can be adjusted by pots U6 and U7.
So you have to decide between the complexity of this circuit and a simpler micro circuit.
I wouldn't worry about the accuracy of the A/D in the micro, since it's just reading a pot.
It's stability you want, and the micro's A/D should be quite stable.