Wierd conspiracy theories could not exist without this handy fallacy, but it is one for which juries are often total suckers.
Argumetium ad ignorantium - "argument from ignorance", or "argument from incredulity", does not mean argument from stupidity - it means argument from a proposition which cannot be proven as fact, or from a proposition which cannot readily be disproven due to "ignorance" - eg lack of data.
The lack of certain evidence for, or against, something does not make it either untrue, or believeable. Nor does the lack of certain evidence for something make an alternative theory more likely to be true.
"It's hard to believe that a couple of terrorists knocked down the Trade Center, so it seems that Bush and the Israelis must have blown up the World Trade Center towers, and faked the TV images with the cooperation of TV news, so Bush could go into Iraq to get free oil to reward his rich friends. It's a huge conspiracy for the benefit of the Illuminati."
Prove that is wrong. It is amazingly easy to make propositions which are simple lies, or absurd, but difficult to prove wrong. Thus you raise doubt, which can appeal to the paranoid part of people.
Here's another: "Abe Lincoln greatly enjoyed his male friends and colleagues, so he probably was gay." Prove that isn't true.
"The authors of the Second Amendment could not have intended that Americans should be killing eachother by having guns to protect their homes, so we should ban guns." Well, name me one of those guy's families which had no guns in their homes, but it's a bit late to do a poll now, so you can assert anything you want, based on your bias. I'd bet every one of the Founders had many guns in their home.
Or, "Would my client, an experienced driver, forget to put n his parking brake? Impossible. My client deliberately declined to put on his parking brake, because he was afraid that the brake might fail on his '57 convertible Chevy, so he put it safely in "park" before it unfortunately, and tragically, but innocently, rolled down over all of those nice families at the beach."
Try to prove I am wrong on that. Doubt? He's a nice guy - your next-door neighbor - not a murderer or a manslaughterer. He just likes antique Chevys, like we all do.