This question is a bit old but I have a different take on it. If you own all counties in a duchy then whether you want to keep that duchy around depends on how many duchy titles you currently own. If three or more, then the duchy is causing you a negative opinion modifier from all your vassals.
In this case destroy the duchy title, don't give it to another character. If you own all counties and give the duchy title to another character, they will get a huge negative opinion modifier unless you give them all of the counties!
If you own two duchies total or less, then keep the duchy until this changes. It's giving you some prestige as has been said, whereas destroying it costs prestige.
I also want to add that this applies only in case that you own all the counties in the relevant duchy. If you don't you will incur -50 opinion with all de jure vassals of the duchy. Same goes for kingdom titles, which can usually most conveniently be disposed of immediately after a crusade.