Day of the week GUI in roblox stays at the same day and does not change?

It's probably a syntax error, try putting parentheses with your not on line 17 like this

while not(game.Lighting.ClockTime == 0) do

I also suggest doing this if you want your script to work after day 7

local daysOfTheWeek = {
    "Sunday",
    "Monday",
    "Tuesday",
    "Wednesday",
    "Thursday",
    "Friday",
    "Saturday"
}

for i=1, #daysOfTheWeek do
    for _, plr in pairs(game.Players:GetPlayers()) do
        local plrGui = plr:WaitForChild("PlayerGui")
        local serverWeekGui = plrGui:WaitForChild("ServerWeekGui")
        serverWeekGui.TextLabel.Text = daysOfTheWeek[i]
    end
    while not game.Lighting.ClockTime == 0 do
        wait(1)
    end

    if i == 7 then
        i = 0
    end
end

Problem:

The problem you're encountering is that you're trying to Wait for the child of "ServerWeekGui". This GUI is a screen GUI, and would not work, because you cannot change the ScreenGui.

Solution

To resolve this, you'd simply have to index the hierarchy of the screen GUI, the (Textabel), and that would fix your error, this would index the text label, so it would actually run!

Fixed Script:

DAY/Night cycle script: (The below script works):

local daysOfTheWeek = {
    "Sunday",
    "Monday",
    "Tuesday",
    "Wednesday",
    "Thursday",
    "Friday",
    "Saturday"
}

for z=1, #daysOfTheWeek do

    for _, plr in pairs(game.Players:GetPlayers()) do
        local plrGui = plr:WaitForChild("PlayerGui")

        local serverWeekGui = plrGui:WaitForChild("ScreenGui").TextLabel1 -- change "TextLabel1" to your textlabel name
        for i = 1, 23 do
            wait(5) -- change how fast the clocktime arises
            game.Lighting.ClockTime = i -- changes the clock time to the index
            if game.Lighting.ClockTime == i then
                serverWeekGui.Text = daysOfTheWeek[z]   -- changes the text to the days of the week
            end
         end
      end
  end