Baby Buttons came to be when I noticed how much my one year-old daughter loved coat and shirt buttons, as well as any sort of electronic buttons. She was the primary beta-tester of Baby Buttons for quite some time, and now her little sister is the new tester!
Baby Buttons offers an interface simple enough for a baby or toddler to use, as it is a series of buttons which provide visual, audible, and physical (iPhone only) feedback when pressed. Give it a shake for an added surprise. Both of my daughters love to play with it, and it keeps the young ones occupied long enough for me, the parent, to finish whatever it is that I'm doing. I hope you will have equal or greater success with it.
Baby Buttons works on old versions of iOS (3.2!) which means you can still run it on the original iPhone or iPod touch. If you're like me, you've probably upgraded your phone over the last few years, and so you can get a nice baby-friendly case to put on your old device and run Baby Buttons for your baby or toddler.