Value change is not detected for some reason?

It seems that your issue is that you forgot to put .Value in the if statements. Besides that, I have some suggestions. First, I would set the BlackoutTime to be a BoolValue, it's a true or false value, rather then using a StringValue. Then, I would use task.wait(), it's more efficient (less throttling, I believe). Lastly, I would use :WaitForChild() with some of the locals. Here are my suggestions: (First script, the one in the part):

local pointlight = script.Parent:WaitForChild("PointLight ")
local LightPart = script.Parent--this doesn't need :WaitForChild() because the script is loaded in after the part (the script is a child of the part).
LightPart.Material = "Neon"
LightPart.BrickColor = BrickColor.new("Institutional white")
local Blackout = workspace:WaitForChild("BlackoutTime") -- I set this to a boolvalue and I used :WaitForChild(""), this makes sure that the value is loaded in before checking for it. This script may load in before the value does.

Blackout.Changed:Connect(function()
	if Blackout.Value == true then
		LightPart.Material = "Plastic"
		LightPart.BrickColor = BrickColor.new("Really black")
		pointlight.Brightness = 0
		pointlight.Range = 0
	elseif Blackout.Value == false then
		LightPart.Material = "Neon"
		LightPart.BrickColor = BrickColor.new("Institutional white")
		pointlight.Brightness = 1
		pointlight.Range = 60
	end
end)

(Second script, the one inside ServerScriptService):

local function BlackoutLoop()
	local BlackoutTime = math.random(2,5) -- 300,600
	local TimeToBlackout = math.random (2,5) -- 900,3000
	local blackoutValue = workspace:WaitForChild("BlackoutTime") -- this will be true or false
	blackoutValue.Value = false
	print("Blackout Ended!")
	task.wait(BlackoutTime) -- task.wait() is better to use, according to what i searched up
	blackoutValue.Value = true
	print("Blackout Started!")
	task.wait(BlackoutTime)
	BlackoutLoop()
end
BlackoutLoop()