In my experience the biggest rock on the road to saving money, whether the person is a child or adult (though I think it's even harder for kids), is short term vs long term gratification. Sometimes it's just so satisfying to walk out of a store with a cute bag in your hand or to taste that chocolate bar right in the moment. Even though we know we could get something "cooler" or "better" down the line, it's difficult to make ourselves wait. We want it now!
So I find a little reward system is helpful to get kids saving money. Give them a sense of accomplishment in the moment - when they choose to put that $5 back in their purse or tuck the $20 from Grandma in the piggy bank rather than run to the toy store. Maybe they can earn "points" based on a weekly check of a savings jar, points that can be redeemed for small non-monetary prizes from Mom and Dad like getting out of a chore or a trip to a park. Keep track of points through an app or a chart that the child has easy access to view. Similarly, perhaps they can "invest" their money in their parents - "Save that $5 for one week and we'll turn it into $10." The point being to get them used to the concept of delayed gratification, even though it may seem contrary to spend money in getting your child to save money!