Sharing player variables through REs doesn't work. Help?

The issue isn't the remote event itself, it's your usage of MouseButton1Up. The player who clicked whatever GuiButton is not the parameter of the event. Instead, the parameters are as follows:

GuiButton.MouseButton1Up:Connect(function(x,y)
	--where x represents the mouse's x position on the screen, and y represents its y position
end)

In fact, you don't even need to get the player that way. Since you're using a LocalScript, you can simply access the LocalPlayer. The truth is, though, that you don't need the LocalPlayer at all. :FireServer() has no required arguments! You may send variable information through this method if necessary, but the player is not required. This is because the server will recognize automatically what client fired the event. As such, OnServerEvent has a default parameter of player, being the player who fired the event.

--Local Script

local replicate = game:GetService("ReplicatedStorage")

script.Parent.MouseButton1Up:Connect(function() --you don't have to enter the parameters if you won't use them
    replicate.speed:FireServer() 
end)

--Server Script

game:GetService("ReplicatedStorage").speed.OnServerEvent:Connect(function(player)
    print(player.Name .. " has bought the thing")
end)

Again, note the fact that player being a parameter of OnServerEvent has nothing to do with the fact that you were sending it through the remote event previously. It is simply the defaut parameter.