Why does my script give me my tools but not the children of the tool?

Here's a nifty solution for you. While I don't know what exactly was wrong with your original code, this modified code will do what you're hoping for.

Essentially, this will rename tools which you own to your tool, and then move them to ServerStorage, transporting them back to you after you respawn. If you need tools to be named something specific and this will interfere with something, I'm sure you can come up with a different method to sift through the tools in ServerStorage - such as a StringValue inside each tool.

I also made it so it will return the tool you are holding as well - which I moved by using FindFirstChildOfClass() inside of Character.

game:GetService("Players").PlayerAdded:Connect(function(player)
	player.CharacterAdded:Connect(function(Character)
	     for _,v in pairs(game.ServerStorage:GetChildren()) do
	        if string.sub(v.Name, 1, string.len(player.Name)) == player.Name then
				v.Parent = player.Backpack
			end
	     end
	
		Character:WaitForChild("Humanoid").Died:Connect(function()
			for _,v in pairs(player.Backpack:GetChildren()) do
				v.Parent = game.ServerStorage
			end
			if Character:FindFirstChildOfClass("Tool") ~= nil then
				Character:FindFirstChildOfClass("Tool").Name = (tostring(player).."'s tool")
				Character:FindFirstChildOfClass("Tool").Parent = game.ServerStorage
			end
		end)
	 end)
	
	player:WaitForChild("Backpack").ChildAdded:Connect(function(tool)
		tool.Name = (tostring(player).."'s tool")
	end)
end)