String Value Not Changing?

You can not store a refference to a value of an instance in a variable. That is why your value is not changing, you are essentially creating a "copy" that is not getting applied to the original instance as you intend.

You would directly need to change the instance's value:

local textLabel = script.Parent.NPCText
local text = script.Parent.Words.Value
local length = string.len(text)
local Option1 = textLabel.Option1
local Option2 = textLabel.Option2
local stringvalue = "Sherman"
local player = textLabel.Parent.Parent.Parent
for stop = 1, length do
    wait(0.1)
    textLabel.Text = string.sub(text, 1, stop)
    Option1.Visible = true
    Option2.Visible = true
end

script.Parent.Words.Changed:Connect(function()
    for no = 1, length do
        wait(.1)
        textLabel.Text = string.sub(text, 1, no)
    end
end)

Option1.MouseButton1Click:Connect(function()
    -- text = "Oh Brilliant! Please Hand Him Over!"
    script.Parent.Words.Value = "Oh Brilliant! Please Hand Him Over!";
    Option1.Visible = false
    Option2.Visible = false
    wait(2)
    textLabel.Parent:Destroy()
end)

Option2.MouseButton1Click:Connect(function()
    Option1.Visible = false
    Option2.Visible = false
    text = "Ah I See, Well Would You Please Search For Him For Me?"
    game.ReplicatedStorage.QuestStatus:FireClient(player,stringvalue)
    game.ServerStorage:FindFirstChild(player.Name.."'s Quests").FindEdgar.Value = 1
    wait(3)
    textLabel.Parent:Destroy()  
end)

If you have any more questions, feel free to ask!

The variable text is not a reference-type, it's a value-type. What this means is that changing won't affect the Text property of Words. In order to change it, you must instead do script.Parent.Words.Text = "Oh Brilliant! Please Hand Him Over!". Ideally you'd also save the text instance Words in a local variable for less indexing.