I imagine there are any number of ways to go about inventing a gag. Perhaps you hear something which sounds ambiguous or absurd, and use that as the basis of the gag. Then with a rough idea of it, work on the punchline and perhaps modify the rest of it to make it work to best effect.
How do people here do it?
Various. Lots of my gags are done as pictures. I guess my favourite methods are noticing a relevant link between two things (e.g. Paul McCartney turning 64 as he gets divorced):
Or just general satire:
It is said that you think of a punch line first then write the sketch to fit
First of all track down Griff's estimable 7 laws of comedy threads.
Sootyj prefers to;
1 Start with a punchline, e.g. political smearing and excrement smearing sound similar, then suppose GordyB does both.
Becomes would Brown smear less if he used better toiler paper?
2 Ask a question and exagerate the answer.
Who's the worst possible person to look after kids?
Becomes; I'm so irresponsible parents choose Fritzell over me as a baby sitter.
There are other tricks.