Why doesn't my script work twice in the same game?

How do I fix my issue?

Use Debounce, this would most probably fix your issue.

What is a debounce?

Debounce is a system of code that prevents something from happening too many times.

How do I use debounce?

It's very simple, just create a variable, let's call it debounce, and set it to false like this:

local debounce = false

Then, right after your MouseClick function, you would need to check if it still set to false, set it to true, and then right after your code is complete set it to false again:

part.MouseClick:Connect(function(player)
if not debounce then -- If debounce is false
debouce = true -- Set debounce to true
-- Do stuff
debounce = false -- set debounce to false
end
end)

Complete code:

local debounce = false
part = script.parent.clicker

part.MouseClick:Connect(function(player)
if not debounce then
debounce = true
local char = player.Character
local hum = char:WaitForChild("Humanoid")
print("Opening the Gui")
print(part)
local PlayerGui = player.PlayerGui

local GuiToBeOpened = PlayerGui:WaitForChild("ssdssdsds")
local clone = game.ServerStorage.LocalScript:Clone()
clone.Parent = GuiToBeOpened["TextButton"..value]
GuiToBeOpened.TextBox.Visible = true
GuiToBeOpened.TextButton1.Visible = true
GuiToBeOpened.TextButton2.Visible = true
GuiToBeOpened.TextButton3.Visible = true
GuiToBeOpened.TextButton4.Visible = true
debounce = false
end
end
end)

But that still didn't explain it to me well enough!

Well I can't give much of a better explanation than that, so check out this link, it goes much more in-depth than me.