I've been thinking of a particular kind of pun, which I can best describe using the quintessential example:

Have you heard? The Pillsbury Doughboy died. [Oh, how sad! How?] Yeast infection.

For want of a better term, I therefore call them "doughboys." The other example I know of is:

Have you heard? Cap'n Crunch died. [Oh, how sad! How?] Cereal killer.

No, I'm not claiming this is a high art form -- but I thought I'd open up the comments for people to add others. Remember, the goal isn't to mock someone (no matter how much they deserve it); it's just a pun, preferably a slightly absurdist one.