You'll find a variety of online games that teach nouns and other parts of speech if you use a search engine. Look for games that are age appropriate and appeal to your children. One example of an online game is called Noun Dunk. Players make baskets with a basketball by identifying a word as a common noun, proper noun or not a noun. If your children are into sports, this may be the perfect online noun game for them.
When you're out and about with your children, have them identify the nouns around you. For example, if you're at the grocery store, ask them to call out nouns they see around them, such as cereal, milk and bread. This is a great time to introduce proper nouns as well. When your children identify children, ask them to identify some proper nouns on the cereal aisle as well. You can do the same thing while driving home from the store, using either the nouns within the car or nouns that your children can see outside.
Write the letters of the alphabet down the side of a piece of paper. Have each child come up with one or more nouns for each letter. Provide specific rules if you'd like to make the game a little more challenging. For example, specify no proper nouns or choose one category of nouns such as things.
For children in the same age range, pick a time limit and have them list as many nouns as possible in that amount of time. You could offer a prize to the winner of this game as an incentive to work quickly. If the children you're working with are farther apart in age, this may not be a fair competition, so consider making the rules tougher for the older child or counting two of his words for one of the younger child's.