We took our son for his first birthday. So obviously they don't "get a lot out of it". They just kind of go with the flow. Slow moving omnimover type rides with lots of bright colors and things to look at are big hits. You should plan to ride Little Mermaid a lot. My son loved it.
The other thing is characters. So at the beginning of the week he was scared of them. But we kept at it and my one clear memory of the trip is on the last night at the Character Spot in Epcot and he was in his pajamas because it was late. Anyway we're in line and he's reaching towards the characters and making whiney noises because he wanted to meet them. So be prepared for the 1 year old to need. some time to get used to the characters.
Also they love balloons so buy a balloon from the vendor on Main Street.