We took our son on his first trip to Disney a week after his first birthday during the first week of May in 2009. He was already walking on that trip and was able to enjoy the fountains at the resort and the play areas in the parks, particularly those with water fountains, e.g. the dig site waterfall in AK, Pooh's playful spot in MK (which is now closed), the splash fountains by test track and between the big fountain and World Showcase at EPCOT and the honey I shrunk the kids playground at HS.
He was in awe on all of the rides and attractions. For us, it wasn't about what he would remember from the trip, but the enriching experiences. There was as much sensory stimulation for him on that trip as in any educational or entertainment book, CD or DVD we had.
I think the key to a pleasant trip with a one year old is to minimize park time. At that age, it is important to be at the park at rope drop, plan a 15-20 minute rest stop at the park's baby care center mid-morning, and be out of there by noon at the latest and stay at the resort for pool time and nap time until at least 5PM. For example, we would go to AK for Monday morning extra magic hours at 8AM, ride the Safari first thing, walk the pagani forest trail, stop at the baby care center for A/C and snack time, do Rafiki's planet watch, and then either FOLK or Flights of Wonder before heading for the exit at 11:30. We would stay at the resort until leaving for EPCOT at a little before 6. At that time, Spaceship Earth, Nemo, Living with the land, Figment, and the Grand Fiesta are all great, slow-moving rides you can take with a lap child that you can walk right on with zero wait time. We would easily ride all of those at a leisurely pace and be on our way out of the park at 8:30 (he would usually fall asleep in the stroller on the walk back to the car).