"Analogy is the Delaware of Cognition"
Solution 1:
Delaware was the first state to ratify the US Constitution, so:
Analogy is the first state of cognition.
(edit to add a bit more info: Delaware as a whole is quite proud of this fact, and "The First State" is one of its official mottos. As an east-coaster, this tidbit sprang immediately to mind for me.)
Solution 2:
I am translating into Italian Surfaces and Essences (which contains that sentence) together with a friend of mine. Since we were not sure about its meaning, my friend asked Hofstadter: his answer was(*) "I used Delaware because it is a peripherical state, with an identity of its own, but left on the fringes. However in the French edition of the book Hofstadter and Sanders settled down for "Finisterrae", to mean a far, far away place.
Of course this does not explain the burst of laughter :-)
(*) my friend reported Hofstadter's answer in Italian, actually; it may even be that he answered in Italian...
Solution 3:
It is hard to say for sure, but a stereotype you hear about Delaware is that it is the "Lower Slower" state. The notion is that the southern part of the state is very rural and therefore the inhabitants are old, unsophisticated, slow-moving, etc.
I think Douglas Hofstadter is attempting to make the point about analogy (ironically using analogy) that it is very primitive or simplistic by comparison with the process of cognition.
The lecture was delivered at Stanford (in Connecticut), which is quite close to Delaware, and it is likely that the audience would be well aware of the Delaware stereotype, thus accounting for their reaction.