What's wrong here?

Problem

Your script is reading line 4 as a line it needs to remember, not as a property that needs to be changed. Just removing the .BrickColor from that line will make the script remember the value as a object, which if you reference it later in the script with .BrickColor afterward. The script will realise that you're intending to change the value of the object.

Solution

while wait(1) do
    math.randomseed(tick())
    local value = math.random(1,8)
        Color = script.Parent --Just remove the .BrickColor and the script will recognize the 'Color' variable as a object.
        if value == 1 then 
            Color.BrickColor = BrickColor.new("Really Red") --Adding the .BrickColor to the end of the variable will make the script realise it is a property needing to change.
        elseif value == 2 then 
            Color.BrickColor = BrickColor.new("Lime green")
        elseif value == 3 then
            Color.BrickColor = BrickColor.new("Hot pink")
        elseif value == 4 then
            Color.BrickColor = BrickColor.new("New Yeller")
        elseif value == 5 then
            Color.BrickColor = BrickColor.new("Toothpaste")
        elseif value == 6 then
            Color.BrickColor = BrickColor.new("Burnt Sienna")
        elseif value == 7 then
            Color.BrickColor = BrickColor.new("Really black")
        elseif value == 8 then
            Color.BrickColor = BrickColor.new("Peral")
    end
end

How your script was reading it before probably was like this. Repeat randomseed: get a value: remember the part's BrickColor for later being Mid gray: if statement: value is equal to two: alright remember the BrickColor Lime Green: wait a second: alright randomseed: get value: remember Lime Green since that's the color of the Brick: value is 5: remember Toothpaste. And it just goes on just remember the changing variable instead of actually changing the property of the object.