How would you get this to remove a string value from a table?

print(script.Name..' loaded')
local players= game.Players

local surviving= { }

roundActive= false
intermission= true

game.Players.PlayerAdded:connect(function(player)
	table.insert(surviving, player.Name)
		print(player.Name..' has joined the game')
		print(table.concat(surviving, ', '))
	player.CharacterAdded:connect(function(character)
		character.Humanoid.Died:connect(function()
			table.remove(surviving, player.Name)
			print(player.Name..' has died')
		end)
	end)
end)

game.Players.PlayerRemoving:connect(function(player)
if surviving:findFirstChild(player.Name) then
	table.remove(surviving, player.Name)
else end
	print(player.Name..' has left')
end)

function getNewList()
local plr= players:GetChildren()
	for i= 1,#plr do
		if surviving:findFirstChild(plr[i].Name) then
			print(plr[i].Name..' is already in the roster')
		else
			table.insert(surviving, plr[i].Name)
				print(plr[i].Name..' added to the roster')
		end
	end
end

function announceSurvivors()
	print(table.concat(surviving, ', '))
end

The FindFirstChild method does not work with arrays(tables).

for key,value in pairs(surviving) do -- the pairs function is not necessary. 
	if player.Name==value then
		--  code
	end
end

The table.remove function removes the value at the index given. To remove a specific value you can loop through the table.

function removeValue(tab, val)
	for o, n in pairs(tab) do
		if val == n then
			table.remove(tab, o)
		end
	end
end

removeValue(surviving, player.Name)