Noah means rest. But why did Lamech, Noah's father, choose the name"Noah". An important step to answering this question is getting a better grasp on what the world was like at Noah's birth (for more info see State of the World: Noah's Birth).
Now he called his name Noah, saying, "This one will give us rest from our work and from the toil of our hands arising from the ground which the LORD has cursed." (Genesis 5:29)Commentators suggest a variety of interpretations for this verse. Some say that Lamech spoke this in a spirit of prophecy. John Gill notes in his commentary that Jarchi suggests that Noah invented the plow (which helped ease man's work). But more can be learned by considering the state of the world at Noah's birth.
The world was growing more wicked. Men were dying (see Natural Death). The task which God had given to men was becoming more and more wearisome. Keeping this in mind, Lamech's choice of name for Noah is appropriate. The work and toil seemed endless. The curse was more apparent than ever. Understandably, the righteous must have longed for rest. They must have longed for someone to soothe them— to save them.
Now faith is the assurance of things hoped for, the conviction of things not seen. For by it the men of old gained approval. (Hebrews 11:1-2)When Lamech said, "This one will give us rest from our work," perhaps he spoke these words in faith. Perhaps Lamech looked down at his newborn son and had faith that Noah would bring rest. What kind of rest did Lamech have in mind? Whatever his thoughts were, the rest which Noah would bring was much greater than Lamech could have ever imagined.
Through Noah, God spared the human race. Through Noah, the planet was repopulated. Through Noah, came Christ. Noah may have brought about some rest, but Christ brings about the real rest. It is only through Christ, and Him alone, that men are able to dwell in the new earth where He will wipe away every tear from their eyes; and there will no longer be any death; and there will not longer be any mourning, or crying, or pain; the first things have passed away (Revelation 21:4).
State of the World (2948 BC): Noah's Birth - Genesis 5:28
State of the World (3130 BC): Noah's Father (Lamech) Is Born - Genesis 5:25
Natural Death - Genesis 5:4-5
Genealogy Timeline - Genesis 5
Psalm 1:2 - Delight in the LORD's Instruction