As a kid growing up I remember having sewing cards with laces. They were fun to string up making sure the lace went through each hole. I have purposely avoided things like this unless they're durable due to how rough my kids are with things. I finally got an idea the other day of how to make one that is more durable yet cheap.
We tend to go through a lot of containers that have recyclable plastic lids from many dairy products. Use a cookie cutter and trace on the lid the shape you want to have. (You might need to enlarge this to some degree but make sure it fits on the lid.) Then cut the shape out and punch holes using a hole punch. If you don't feel like cutting it out a circle will do just fine although it will have a rim. For safety you might consider using electrical tape and going around the edges to prevent any sharp edges.
To string the lacing card you can use a shoelace, pipecleaner, a wet spaghetti noodle, or yarn just to name a few things.
The only word of caution I will give you is that make sure there are no sharp edges that your child might scratch or cut themselves on.