I bought an Archos 101 Android based tablet ($294 on Amazon) for my daughter. A touchscreen interface is very intuitive for a 2 year old. Before she always pointed to the Youtube related video on my computer and say "How bout this one?" and I would use the mouse and click on it. Now she can just touch the video she wants.
It's more of a toy/learning experience for me, but it is great for her to keep her occupied. I have many kid apps like drawing, connect the dots, memory game, color Mickey Mouse, etc. on it. I have all of her favorite YouTube videos downloaded to it along with some Disney movies and videos of her. She can also finger scroll through all the pictures we took of her which will keep her entertained for awhile. Great for at home, in the car or restaurants, or on a plane.
It's not perfect as she is still learning the interface and accidentally presses the home button or a link in an app that starts up the web browser to rate it or to email, but generally it works out quite well especially that the battery life is about 8 hours.