Why can't I find the player name?

its because the script ran before the character actually loaded, making the Character property of Player nil. instead you can use CharacterAdded which listens for when the player's character loaded or respawned. also use local functions and :connect() is deprecated, use :Connect().

local name

local function onPlayerEntered(player)
    player.CharacterAdded:Wait() -- yield until the character loaded or instead of using :Wait() you could make an anonymous function

    name = player.Character.Name
    print(name)

end

game.Players.PlayerAdded:Connect(onPlayerEntered)

local name

function onPlayerEntered(player)
    name = player:WaitForChild("Character").Name
    print(name)

end

game.Players.PlayerAdded:connect(onPlayerEntered)

It takes some time to Load the Character in

The gist of things

The issue here is that the player and the Character load at different times, as the player loads first and the Character and all scripts , guis, and tools inside the player load after. 

Unfortunately, the character loads slower than the script runs, which means you would be getting the character property of the player without it existing. That is the reason why that line of code errored.

Solutions

There are a couple of solutions here, first, you can simply not use the name of the character as the names of the player and the character should be identical.

game:GetService("Players").PlayerAdded:Connect(function(plr)
    print(plr.Name)
end)

The second solution would be to use the character added event, which there are two ways of utilizing. The ``RBXScriptSignal`` object has two functions of it that can be used here, the ``Wait()`` functions and the ``Connect()`` function. The wait function yields until the signal is fired, and returns a tuple of arguments

game:GetService("Players").PlayerAdded:Connect(function(plr)
    local char = plr.CharacterAdded:Wait()
    print(char.Name)
end)

This method is not recommended as it holds up all other parts of the PlayerAdded event (its current thread) as it waits for the event to be fired

A far better way to do this would be to use the :Connect() function of the RBXScriptSignal

game:GetService("Players").PlayerAdded:Connect(function(plr)
    plr.CharacterAdded:Connect(function(char)
        print(char.Name)
    end)
end)

Note: the plr.CharacterAdded:Wait() method is not used commonly but the plr.CharacterAdded:Wait() part it self can be useful outside the player added event, for example, in local scripts outside the Character, many people simply do

local plr = game.Players.LocalPlayer
local char = plr.Character or plr.CharacterAdded:Wait()

to make sure the character exist before using it

Hopefully this helped and have a nice day scripting!