What does Genesis 2:19 mean?
This verse reveals another aspect of man's purpose in the world God had made. The man was tasked with using his God-given authority and creativity in order to name the animals. The picture painted is powerful, empowering, and sweet, in a way. We see God forming the wild animals and birds, bringing them into existence, and then, eventually, bringing them to the man to discover what the man would call them.God seems to be taking pleasure watching the man accomplish this task. It appears that God is not directing the naming of the animals in any way; He is truly leaving it to the man to use his own creativity, judgment, and process to come up with these names. And then God allows those names to stand as the animals' true names. It is truly a privilege and honor which God bestows on this man, by allowing him to participate in the work of building and maintaining this new creation.
More than that, the act of naming something is meaningful in the book of Genesis. This act often implies rule over and responsibility for that thing. God has already instructed man to subdue the earth and rule over all of the creatures (Genesis 1:28). Having the man name the animals is another way of giving him responsibility to rule, subdue, and care for the animals.
This passage is also important in the context of verse 18, where God stated His intent to make a helper for the man. By bringing all the animals in person, God can emphasize the fact that none of them are suitable partners. The man will be able to see, first-hand that no other created being is his equal. Only other people, male and female, are meant to be joined as "one flesh" (Genesis 2:23) and share that level of connection.