He quite excitedly told Einstein and Morgenstern that he had found a weakness in the Constitution that would allow the US to become a dictatorship. They were horrified and tried to tell him not to bring up such ideas at the hearing, but the judge happened to note that it was a good thing that in the US, unlike Gödel's native Austria, we were not a dictatorship -- which prompted Gödel to begin explaining his discovery...[2]
Unfortunately, no one ever bothered to write down what Gödel's idea was! But it has been speculated[3] that it is this: that the Article V provision for constitutional amendments does not exclude that provision itself from amendment -- so it is possible to first pass an amendment making it easier to pass amendments, after which it could be a short trip to dictatorship under the right conditions.
Fanciful, but a fascinating story. Picturing Gödel, Einstein, and Morgenstern hanging out and being playful with each other is a treat.
[1] http://morgenstern.jeffreykegler.com
[2] https://d78508e2-a-97b1dc77-s-sites.googlegroups.com/a/jeffr...
[3] http://poseidon01.ssrn.com/delivery.php?ID=42406910501309506...