I have a couple. Awesome for practical usage. Point and shoot, nothing complicated about it, it has like two or three buttons on it. The quality is pretty good, however, you HAVE to shoot when there's a lot of light. At night it sucks unless you're right under a light, or in a dimly lit area it kinda blows.
I got one for Christmas. I thought it worked extremely well, especially in low light indoors. I let my kid use it because it's pretty durable and easy to use and upload to youtube, etc.
I wouldn't say it's a substitute for a video camera, because it lacks a zoom, but I love how it's a little bigger than pack of cards and boots up real quick, so it's perfect for your situation with a one year old.
It's not like you're going to be taking shots from the bleachers in some little league game yet. And at that age, stuff happens really fast.
Just promise to post the video when you get smacked in the nutz. Those videos are so HILARIOUZZZZZZZZZZZZZZZZZZZZZZZ.