How to define a players name in a variable?

You can create the variable on PlayerAdded, like this:

local Players = game.Players

Players.PlayerAdded:Connect(function(playerWhoJoined)
	_G.LastPlayerToJoinGame = playerWhoJoined.Name
end)

Obviously, this is gonna get overwritten each time someone new joins, so we'll need to build onto this a bit if you wanna store the names of everyone who joins. However if that's the case you're better off just keeping what you need in your PlayerAdded, and passing the player as an argument to other scripts through events for other things you may need.