I used Crayola Model Magic (affiliate link) to make this. It comes in different colors and is sold in the Crayola section at many stores. It is very easy to work with and best of all- it's not even the least bit messy! It will get hard but still be bendable - it doesn't get a solid hard like regular clay does. If you accidentally drop it on the floor, it won't shatter.
Roll a portion of the clay into a ball. Flatten it out with the palm of your hand on a smooth surface. Keep turning it over so that both sides are smooth. You can either flatten it into a circle shape or if you wanted a perfect circle shape, you could find a big enough circle shape around the house that you could press into the clay. Next, firmly press your child's hand into the clay (if you mess up, just start over). Then, I took a fork and pressed it along the edge to make a fun border. I used one of those miniature screwdrivers to press 2 holes into it so it could be hung up with ribbon. I also used the screwdriver to inscribe the date- you could also put the child's age instead.
Click HERE to see another example I made of my son's hand and foot.