Is there a way to make a Click Detector cooldown?

Yes. You're on the right track, you're just not utilizing the debounce correctly.

local part = script.Parent 
local debounce = false

part.ClickDetector.MouseClick:Connect(function() 
    if not debounce then -- This is the check we need.
        -- This will check if "debounce" is equal to false; when it is false, run this code, and we change it to true so that it won't activate while on cooldown.
        part.Sign.SurfaceGui.TextLabel.Text = math.random(1, 8) 
        debounce = true 
        wait(6) 
        debounce = false
    end
end)

Maybe something like this(Not sure):

CoolDown = false

function Example()

CoolDown = true

part.Sign.SurfaceGui.TextLabel.Text = math.random(1, 8) 

Wait(3)

CoolDown = false

end

function Clicked()

if CoolDown == false then

    Example()

end

end