When I was little, I was completely captivated by Sea World. It's a little expensive (although your baby will be free), so I'd only go if the adults in the party would like it too, but I really did enjoy it at that age. The pictures are hilarious because I am always completely wide-eyed, taking everything in!
Tidepooling at La Jolla Cove would be fun with a one-year-old. Just the beach in general, too!