Favorite thing: When I was in Dubrovnik in both February and in March (of different years) there weren't many islands that one was able to go to on a daytrip. By default, I ended up on Mljet, which is only a twohour hydrofoil ride away. It was a really beautiful island (reminded me of Jurassic Park without the dinos) but impossible to get around that time of the year without renting a car (standards only) or hitching. I ended up hitching. Go to the park, because that's the "must see" of the island. But pack a snack with you. The only restaurant open was one that was 4 kilos from the ferry dock.
Fondest memory: There are some rocky islands close to the coast of Dubrovnik. Lokrum island (see photo) is the closest from Grad, the old city.